Mathematics · Class 11 Science
Ch 9Differential Calculus – Limits and Continuity — Class 11 Mathematics, concept-first.
Calculus is fundamentally the mathematics of change, and its reach extends across virtually every branch of science and social science. Whenever an observer wants to know not just the value of a changing quantity but how fast it is changing — the rate of change — calculus is the tool required, because most changing qua…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Concept of Limit
A limit answers one question: as the input crowds in on some point from both sides, what value does crowd in on? Crucially, this is a statement about the neighbourhood of , not about the point itself.
Most relevant Q&A
- Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to2}\dfrac{x-2}{x^2-x-2}$$ | $x$ | $1.9$ | $1.99$…Free
- Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to2}\dfrac{x-2}{x^2-4}$$ | $x$ | $1.9$ | $1.99$ |…Free
- Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to0}\dfrac{\sqrt{x+3}-\sqrt3}{x}$$ | $x$ | $-0.1$…Free
- Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to-3}\dfrac{\sqrt{1-x}-2}{x+3}$$ | $x$ | $-3.1$ |…Preview
- Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to0}\dfrac{\sin x}{x}$$ | $x$ | $-0.1$ | $-0.01$ |…Preview
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
Calculus is fundamentally the mathematics of change, and its reach extends across virtually every branch of science and social science.
Limits
This section develops the notion of a limit carefully, moving from an intuitive, table-and-graph based understanding towards a small set of theorems that let limits be computed mechanically rather tha…
The calculation of limits
The idea of a limit asks a very specific question: as the input of a function is pushed closer and closer to some fixed value (without ever actually landing on ), what value does the output get closer…
One sided limits
Several of the illustrations in the previous section relied informally on the idea of approaching a point from just one side; this section makes that idea precise.
+−Exercise 9.1i23 questions
- Q1Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to2}\dfrac{x-2}{x^2-x-2}$$ | $x$ | $1.9$ | $1.99$…Free
- Q2Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to2}\dfrac{x-2}{x^2-4}$$ | $x$ | $1.9$ | $1.99$ |…Free
- Q3Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to0}\dfrac{\sqrt{x+3}-\sqrt3}{x}$$ | $x$ | $-0.1$…Free
- Q4Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to-3}\dfrac{\sqrt{1-x}-2}{x+3}$$ | $x$ | $-3.1$ |…Preview
- Q5Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to0}\dfrac{\sin x}{x}$$ | $x$ | $-0.1$ | $-0.01$ |…Preview
- Q6Complete the table using a calculator and use the result to estimate the limit. $$\lim_{x\to0}\dfrac{\cos x-1}{x}$$ | $x$ | $-0.1$ | $-0.01$…Preview
- Q7Use the graph (Fig. 9.13) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to3}(4-x)$$ ![Graph of the st…Preview
- Q8Use the graph (Fig. 9.14) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to1}(x^2+2)$$ ![Graph of the…Preview
- Q9Use the graph (Fig. 9.15) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to2}f(x),\qquad \text{where }…Preview
- Q10Use the graph (Fig. 9.16) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to1}f(x),\qquad \text{where }…Preview
- Q11Use the graph (Fig. 9.17) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to3}\dfrac1{x-3}$$ ![Graph of…Preview
- Q12Use the graph (Fig. 9.18) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to5}\dfrac{|x-5|}{x-5}$$ ![Gr…Preview
- Q13Use the graph (Fig. 9.19) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to1}\sin(\pi x)$$ ![Graph of…Preview
- Q14Use the graph (Fig. 9.20) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to0}\sec x$$ ![Graph of y = s…Preview
- Q15Use the graph (Fig. 9.21) to find the limit, if it exists. If the limit does not exist, explain why. $$\lim_{x\to\pi/2}\tan x$$ ![Graph of y…Preview
- Q16Sketch the graph of $f$, then identify the values of $x_0$ for which $\displaystyle\lim_{x\to x_0}f(x)$ exists. $$f(x)=\begin{cases}x^2, & x…Preview
- Q17Sketch the graph of $f$, then identify the values of $x_0$ for which $\displaystyle\lim_{x\to x_0}f(x)$ exists. $$f(x)=\begin{cases}\sin x,…Preview
- Q18Sketch the graph of a function $f$ that satisfies the given values. **(i)** $f(0)$ is undefined $\displaystyle\lim_{x\to0}f(x)=4$ $f(2)=6$ $…Preview
- Q19Write a brief description of the meaning of the notation $\displaystyle\lim_{x\to8}f(x)=25$.Preview
- Q20If $f(2)=4$, can you conclude anything about the limit of $f(x)$ as $x$ approaches $2$?Preview
- Q21If the limit of $f(x)$ as $x$ approaches $2$ is $4$, can you conclude anything about $f(2)$? Explain your reasoning.Preview
- Q22Evaluate $\displaystyle\lim_{x\to3}\dfrac{x^2-9}{x-3}$, if it exists, by finding $f(3^-)$ and $f(3^+)$.Preview
- Q23Verify the existence of $\displaystyle\lim_{x\to1}f(x)$, where $$f(x)=\begin{cases}\dfrac{|x-1|}{x-1}, & x\ne1\\ 0, & x=1\end{cases}$$Preview
Theorems on limits
Building a table of values or sketching a graph every time a limit needs to be found is neither practical nor rigorous.
+−Exercise 9.2i15 questions
- Q1Evaluate the following limit: $$\lim_{x\to2}\dfrac{x^4-16}{x-2}$$Free
- Q2Evaluate the following limit ($m$ and $n$ are integers): $$\lim_{x\to1}\dfrac{x^m-1}{x^n-1}$$Free
- Q3Evaluate the following limit: $$\lim_{\sqrt x\to3}\dfrac{x^2-81}{\sqrt x-3}$$Free
- Q4Evaluate the following limit ($x>0$): $$\lim_{h\to0}\dfrac{\sqrt{x+h}-\sqrt x}{h}$$Preview
- Q5Evaluate the following limit: $$\lim_{x\to5}\dfrac{\sqrt{x+4}-3}{x-5}$$Preview
- Q6Evaluate the following limit: $$\lim_{x\to2}\dfrac{\dfrac1x-\dfrac12}{x-2}$$Preview
- Q7Evaluate the following limit: $$\lim_{x\to1}\dfrac{\sqrt x-x^2}{1-\sqrt x}$$Preview
- Q8Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt{x^2+1}-1}{\sqrt{x^2+16}-4}$$Preview
- Q9Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt{1+x}-1}{x}$$Preview
- Q10Evaluate the following limit: $$\lim_{x\to1}\dfrac{\sqrt[3]{7+x^3}-\sqrt{3+x^2}}{x-1}$$Preview
- Q11Evaluate the following limit: $$\lim_{x\to2}\dfrac{2-\sqrt{x+2}}{\sqrt[3]2-\sqrt[3]{4-x}}$$Preview
- Q12Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt{1+x^2}-1}{x}$$Preview
- Q13Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt{1-x}-1}{x^2}$$Preview
- Q14Evaluate the following limit: $$\lim_{x\to5}\dfrac{\sqrt{x-1}-2}{x-5}$$Preview
- Q15Evaluate the following limit ($a>b$): $$\lim_{x\to a}\dfrac{\sqrt{x-b}-\sqrt{a-b}}{x^2-a^2}$$Preview
Infinite limits and limits at infinity
Not every limit settles down to a finite real number as approaches a point — sometimes the function values grow without any bound at all.
Limits at infinity
Section 9.2.4 let approach a finite point and asked what happens to when becomes unbounded. This section reverses the roles: now itself is allowed to grow without bound (positively or negatively), and…
Limits of rational functions
For a rational function , the technique of Section 9.2.5 (divide by the highest power of in the denominator) can be packaged into a single rule of thumb that avoids repeating the division every time,…
Applications of limits
Limits at infinity and one-sided limits are not merely abstract exercises — they routinely answer a natural real-world question: "what is the extreme (maximum or minimum, long-run or boundary) behavio…
+−Exercise 9.3i10 questions
- Q1**(a)** Find the left and right limits of $f(x)=\dfrac{x^2-4}{(x^2+4x+4)(x+3)}$ at $x=-2$. **(b)** $f(x)=\tan x$ at $x=\dfrac{\pi}{2}$.Free
- Q2Evaluate the following limit: $$\lim_{x\to3}\dfrac{x^2-9}{x^2(x^2-6x+9)}$$Free
- Q3Evaluate the following limit: $$\lim_{x\to\infty}\left(\dfrac3{x-2}-\dfrac{2x+11}{x^2+x-6}\right)$$Free
- Q4Evaluate the following limit: $$\lim_{x\to\infty}\dfrac{x^3+x}{x^4-3x^2+1}$$Preview
- Q5Evaluate the following limit: $$\lim_{x\to\infty}\dfrac{x^4-5x}{x^2-3x+1}$$Preview
- Q6Evaluate the following limit: $$\lim_{x\to\infty}\dfrac{1+x-3x^3}{1+x^2+3x^3}$$Preview
- Q7Evaluate the following limit: $$\lim_{x\to\infty}\left(\dfrac{x^3}{2x^2-1}-\dfrac{x^2}{2x+1}\right)$$Preview
- Q8Show that: **(i)** $\displaystyle\lim_{n\to\infty}\dfrac{1+2+3+\cdots+n}{3n^2+7n+2}=\dfrac16$ **(ii)** $\displaystyle\lim_{n\to\infty}\dfrac…Preview
- Q9An important problem in fishery science is to estimate the number of fish presently spawning in streams and use this information to predict…Preview
- Q10A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tan…Preview
Sandwich Theorem
Some limits cannot be evaluated by any of the algebra-of-limits theorems directly — typically because the expression involves a factor, such as , that oscillates and has no limit of its own.
Two special Trigonometrical limits
Result 9.1. (a) (b) .
Some important other limits
Result 9.2.
+−Exercise 9.4i28 questions
- Q1Evaluate the following limit: $$\lim_{x\to\infty}\left(1+\dfrac1x\right)^{7x}$$Free
- Q2Evaluate the following limit: $$\lim_{x\to0}(1+x)^{1/(3x)}$$Free
- Q3Evaluate the following limit: $$\lim_{x\to\infty}\left(1+\dfrac kx\right)^{m/x}$$Free
- Q4Evaluate the following limit: $$\lim_{x\to\infty}\left(\dfrac{2x^2+3}{2x^2+5}\right)^{8x^2+3}$$Preview
- Q5Evaluate the following limit: $$\lim_{x\to\infty}\left(1+\dfrac3x\right)^{x+2}$$Preview
- Q6Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sin^3(x/2)}{x^3}$$Preview
- Q7Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sin\alpha x}{\sin\beta x}$$Preview
- Q8Evaluate the following limit: $$\lim_{x\to0}\dfrac{\tan2x}{\sin5x}$$Preview
- Q9Evaluate the following limit: $$\lim_{\alpha\to0}\dfrac{\sin(\alpha^n)}{(\sin\alpha)^m}$$Preview
- Q10Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sin(a+x)-\sin(a-x)}{x}$$Preview
- Q11Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt{x^2+a^2}-a}{\sqrt{x^2+b^2}-b}$$Preview
- Q12Evaluate the following limit: $$\lim_{x\to0}\dfrac{2\arcsin x}{3x}$$Preview
- Q13Evaluate the following limit: $$\lim_{x\to0}\dfrac{1-\cos x}{x^2}$$Preview
- Q14Evaluate the following limit: $$\lim_{x\to0}\dfrac{\tan2x}{x}$$Preview
- Q15Evaluate the following limit: $$\lim_{x\to0}\dfrac{2^x-3^x}{x}$$Preview
- Q16Evaluate the following limit: $$\lim_{x\to0}\dfrac{3^x-1}{\sqrt{x+1}-1}$$Preview
- Q17Evaluate the following limit: $$\lim_{x\to0}\dfrac{1-\cos^2x}{x\sin2x}$$Preview
- Q18Evaluate the following limit: $$\lim_{x\to\infty}x\left[3^{1/x}+1-\cos\!\left(\dfrac1x\right)-e^{1/x}\right]$$Preview
- Q19Evaluate the following limit: $$\lim_{x\to\infty}\big\{x[\log(x+a)-\log(x)]\big\}$$Preview
- Q20Evaluate the following limit: $$\lim_{x\to\pi}\dfrac{\sin3x}{\sin2x}$$Preview
- Q21Evaluate the following limit: $$\lim_{x\to\pi/2}(1+\sin x)^{2\csc x}$$Preview
- Q22Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt2-\sqrt{1+\cos x}}{\sin^2x}$$Preview
- Q23Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{\tan x}$$Preview
- Q24Evaluate the following limit: $$\lim_{x\to\infty}\left(\dfrac{x^2-2x+1}{x^2-4x+2}\right)^{x}$$Preview
- Q25Evaluate the following limit: $$\lim_{x\to0}\dfrac{e^x-e^{-x}}{\sin x}$$Preview
- Q26Evaluate the following limit: $$\lim_{x\to0}\dfrac{e^{ax}-e^{bx}}{x}$$Preview
- Q27Evaluate the following limit: $$\lim_{x\to0}\dfrac{\sin x(1-\cos x)}{x^3}$$Preview
- Q28Evaluate the following limit: $$\lim_{x\to0}\dfrac{\tan x-\sin x}{x^3}$$Preview
Continuity
Continuity is the mathematical way of capturing an everyday intuition — that many natural processes change smoothly, without sudden jumps: a rod expanding continuously as it is heated, an organism gro…
Examples of functions Continuous at a point
At every point of their domain, the following standard families of functions are continuous — each fact follows directly from the algebra-of-limits theorems (Section 9.2.3) applied at a general point…
Algebra of continuous functions
Just as limits combine algebraically (Theorem 9.2), so does continuity — which should be unsurprising, since continuity is itself defined directly in terms of limits.
Removable and Jump Discontinuities
40 QNot all discontinuities are the same "kind" — some can be patched by simply reassigning a single function value, while others cannot be patched at all no matter what value is chosen.
+−Exercise 9.5i15 questions
- Q1Prove that $f(x)=2x^2+3x-5$ is continuous at all points in $\mathbb R$.Free
- Q2Examine the continuity of the following: **(i)** $x+\sin x$ **(ii)** $x^2\cos x$ **(iii)** $e^x\tan x$ **(iv)** $e^{2x}+x^2$ **(v)** $x\ln x…Free
- Q3Find the points of discontinuity of the function $f$, where **(i)** $f(x)=\begin{cases}4x+5, & x\le3\\ 4x-5, & x>3\end{cases}$ **(ii)** $f(x…Free
- Q4At the given point $x_0$ discover whether the given function is continuous or discontinuous, citing reasons for your answer: **(i)** $x_0=1,…Preview
- Q5Show that the function $$f(x)=\begin{cases}\dfrac{x^3-1}{x-1}, & x\ne1\\ 3, & x=1\end{cases}$$ is continuous on $(-\infty,\infty)$.Preview
- Q6For what value of $\alpha$ is the function $$f(x)=\begin{cases}\dfrac{x^4-1}{x-1}, & x\ne1\\ \alpha, & x=1\end{cases}$$ continuous at $x=1$?Preview
- Q7Let $$f(x)=\begin{cases}0, & x<0\\ x^2, & 0\le x<2\\ 4, & x\ge2\end{cases}$$ Graph the function. Show that $f(x)$ is continuous on $(-\infty…Preview
- Q8If $f$ and $g$ are continuous functions with $f(3)=5$ and $\displaystyle\lim_{x\to3}[2f(x)-g(x)]=4$, find $g(3)$.Preview
- Q9Find the points at which $f$ is discontinuous. At which of these points is $f$ continuous from the right, from the left, or neither? Sketch…Preview
- Q10A function $f$ is defined as follows: $$f(x)=\begin{cases}0, & x<0\\ x, & 0\le x<1\\ -x^2+4x-2, & 1\le x<3\\ 4-x, & x\ge3\end{cases}$$ Is th…Preview
- Q11Which of the following functions $f$ has a removable discontinuity at $x=x_0$? If the discontinuity is removable, find a function $g$ that a…Preview
- Q12Find the constant $b$ that makes $g$ continuous on $(-\infty,\infty)$: $$g(x)=\begin{cases}x^2-b^2, & x<4\\ bx+20, & x\ge4\end{cases}$$Preview
- Q13Consider the function $f(x)=x\sin\dfrac\pi x$. What value must we give $f(0)$ in order to make the function continuous everywhere?Preview
- Q14The function $f(x)=\dfrac{x^2-1}{x^3-1}$ is not defined at $x=1$. What value must we give $f(1)$ in order to make $f(x)$ continuous at $x=1$…Preview
- Q15State how continuity is destroyed at $x=x_0$ for each of the following graphs (Fig. 9.38-9.41). **(a)** A curve drawn for $x<x_0$ ends at a…Preview
+−Exercise 9.6i25 questions
- Q1$\displaystyle\lim_{x\to\infty}\dfrac{\sin x}{x}$ (1) $1$ (2) $0$ (3) $\infty$ (4) $-\infty$Free
- Q2$\displaystyle\lim_{x\to\pi/2}\dfrac{2x-\pi}{\cos x}$ (1) $2$ (2) $1$ (3) $-2$ (4) $0$Free
- Q3$\displaystyle\lim_{x\to0}\dfrac{\sqrt{1-\cos2x}}{x}$ (1) $0$ (2) $1$ (3) $\sqrt2$ (4) $\text{does not exist}$Free
- Q4$\displaystyle\lim_{\theta\to0}\dfrac{\sin\sqrt\theta}{\sqrt{\sin\theta}}$ (1) $1$ (2) $-1$ (3) $0$ (4) $2$Preview
- Q5$\displaystyle\lim_{x\to\infty}\left(\dfrac{x^2+5x+3}{x^2+x+3}\right)^{x}$ is (1) $e^4$ (2) $e^2$ (3) $e^3$ (4) $1$Preview
- Q6$\displaystyle\lim_{x\to\infty}\dfrac{\sqrt{x^2-1}}{2x+1}=$ (1) $1$ (2) $0$ (3) $-1$ (4) $\dfrac12$Preview
- Q7$\displaystyle\lim_{x\to\infty}\dfrac{a^x-b^x}{x}=$ (1) $\log ab$ (2) $\log\!\left(\dfrac ab\right)$ (3) $\log\!\left(\dfrac ba\right)$ (4)…Preview
- Q8$\displaystyle\lim_{x\to0}\dfrac{8^x-4^x-2^x+1}{x^2}=$ (1) $2\log2$ (2) $2(\log2)^2$ (3) $\log2$ (4) $3\log2$Preview
- Q9If $f(x)=x(-1)^{\left\lfloor\frac1x\right\rfloor},\ x\le0$, then the value of $\displaystyle\lim_{x\to0}f(x)$ is equal to (1) $-1$ (2) $0$ (…Preview
- Q10$\displaystyle\lim_{x\to3}\lfloor x\rfloor=$ (1) $2$ (2) $3$ (3) $\text{does not exist}$ (4) $0$Preview
- Q11Let the function $f$ be defined by $f(x)=\begin{cases}3x, & 0\le x\le1\\ -3x+5, & 1<x\le2\end{cases}$, then (1) $$\displaystyle\lim_{x\to1}f…Preview
- Q12If $f:\mathbb R\to\mathbb R$ is defined by $f(x)=\lfloor x-3\rfloor+|x-4|$ for $x\in\mathbb R$, then $\displaystyle\lim_{x\to3^-}f(x)$ is eq…Preview
- Q13$\displaystyle\lim_{x\to0}\dfrac{xe^x-\sin x}{x}$ is (1) $1$ (2) $2$ (3) $3$ (4) $0$Preview
- Q14If $\displaystyle\lim_{x\to0}\dfrac{\sin px}{\tan3x}=4$, then the value of $p$ is (1) $6$ (2) $9$ (3) $12$ (4) $4$Preview
- Q15$\displaystyle\lim_{\alpha\to\pi/4}\dfrac{\sin\alpha-\cos\alpha}{\alpha-\pi/4}$ is (1) $\sqrt2$ (2) $\dfrac1{\sqrt2}$ (3) $1$ (4) $2$Preview
- Q16$\displaystyle\lim_{n\to\infty}\left(\dfrac1{n^2}+\dfrac2{n^2}+\dfrac3{n^2}+\cdots+\dfrac n{n^2}\right)$ is (1) $\dfrac12$ (2) $0$ (3) $1$ (…Preview
- Q17$\displaystyle\lim_{x\to0}\dfrac{e^{\sin x}-1}{x}=$ (1) $1$ (2) $e$ (3) $\dfrac1e$ (4) $0$Preview
- Q18$\displaystyle\lim_{x\to0}\dfrac{e^{\tan x}-e^x}{\tan x-x}=$ (1) $1$ (2) $e$ (3) $\dfrac12$ (4) $0$Preview
- Q19The value of $\displaystyle\lim_{x\to0}\dfrac{\sin x}{\sqrt{x^2}}$ is (1) $1$ (2) $-1$ (3) $0$ (4) $\infty$Preview
- Q20The value of $\displaystyle\lim_{x\to k^-}\big(x-\lfloor x\rfloor\big)$, where $k$ is an integer, is (1) $-1$ (2) $1$ (3) $0$ (4) $2$Preview
- Q21At $x=\dfrac32$ the function $f(x)=\dfrac{|2x-3|}{2x-3}$ is (1) $\text{continuous}$ (2) $\text{discontinuous}$ (3) $\text{differentiable}$ (…Preview
- Q22Let $f:\mathbb R\to\mathbb R$ be defined by $f(x)=\begin{cases}x, & x\text{ is irrational}\\ 1-x, & x\text{ is rational}\end{cases}$, then $…Preview
- Q23The function $f(x)=\begin{cases}\dfrac{x^2-1}{x^3+1}, & x\ne-1\\ P, & x=-1\end{cases}$ is not defined for $x=-1$. The value of $f(-1)$ so th…Preview
- Q24Let $f$ be a continuous function on $[2,5]$. If $f$ takes only rational values for all $x$ and $f(3)=12$, then $f(4.5)$ is equal to (1) $\df…Preview
- Q25Let a function $f$ be defined by $f(x)=\dfrac{x-|x|}{x}$ for $x\ne0$ and $f(0)=2$. Then $f$ is (1) $\text{continuous nowhere}$ (2) $\text{co…Preview
Summary
This chapter built the idea of a limit from the ground up and used it to define continuity.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 28 questionsHide questions28 questions
- Q1The function $f(x) = \tan x$ is continuous in: (a) $\left[\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right]$ (b) $(-\infty, \infty)$ (c) $\left(\dfrac{…Preview
- Q2$\displaystyle\lim_{x\to 0^-} \dfrac{3x+|x|}{6x+|x|}$ is: (a) $1$ (b) $\dfrac{1}{2}$ (c) $\dfrac{2}{5}$ (d) $\dfrac{5}{2}$Preview
- Q3Verify the continuity at the point $x=0$ for the function $f(x) = \begin{cases} \dfrac{\sin 3x}{x} + 1, & x \ne 0 \\ 2, & x = 0 \end{cases}$…Preview
- Q4For the function $f(x)=\begin{cases}x+2, & x>0\\ x-2, & x<0\end{cases}$ (a) $\lim\limits_{x\to 2^-} f(x)=-1$ (b) $\lim\limits_{x\to 0} f(x)$…Preview
- Q5Define a continuous function on the closed interval $[a, b]$.Preview
- Q6Consider the function $f(x)=\sqrt{x}, x\ge 0$. Does $\lim\limits_{x\to 0} f(x)$ exist?Preview
- Q7Examine the continuity of the function $\cot x+\tan x$.Preview
- Q8$\displaystyle\lim_{\theta \to 0} \dfrac{\sin\sqrt{\theta}}{\sqrt{\sin\theta}}$: (a) 1 (b) -1 (c) 0 (d) 2Preview
- Q9Let $f: \mathbf{R} \to \mathbf{R}$ be defined by $f(x) = \begin{cases} x, & x \text{ is irrational} \\ 1-x, & x \text{ is rational} \end{cas…Preview
- Q10Evaluate: $\displaystyle\lim_{x \to 1} \dfrac{(x + x^2 + x^3 + \ldots + x^n) - n}{x - 1}$Preview
- Q11Do the limits of following functions exist as $x \to 0$? State reasons for your answer. $\dfrac{\sin(x - \lfloor x \rfloor)}{x - \lfloor x \…Preview
- Q12(a) Prove that: $\displaystyle\lim_{\theta \to 0} \dfrac{\sin\theta}{\theta} = 1$ **OR** (b) If $x = a(\theta + \sin\theta)$, $y = a(1-\cos\…Preview
- Q13At $x = \frac{3}{2}$ the function $f(x) = \frac{|2x-3|}{2x-3}$ is: (a) differentiable (b) continuous (c) non-zero (d) discontinuousPreview
- Q14$\lim_{x\to\infty} \left(\dfrac{x^2+5x+3}{x^2+x+3}\right)^x$ is: (a) $e^3$ (b) $e^4$ (c) $1$ (d) $e^2$Preview
- Q15$\displaystyle\lim_{x\to 0} \frac{\sqrt{1-\cos 2x}}{x} =$ (a) 1 (b) $\sqrt{2}$ (c) 0 (d) None of the abovePreview
- Q16Evaluate the limit $\displaystyle\lim_{\sqrt{x}\to 3} \frac{x^2-81}{\sqrt{x}-3}$.Preview
- Q17Evaluate: $\displaystyle\lim_{n\to\infty}\left[6^n + 5^n\right]^{\frac{1}{n}}$.Preview
- Q18Find the constant $b$ that makes $g$ continuous on $(-\infty,\infty)$: $g(x) = \begin{cases}x^2-b^2 & \text{if } x<4\\ bx+20 & \text{if } x\…Preview
- Q19$\displaystyle\lim_{x\to\infty}\dfrac{\sin x}{x}$ (a) $\infty$ (b) $1$ (c) $-\infty$ (d) $0$Preview
- Q20$\displaystyle\lim_{x\to 0}\dfrac{a^x-b^x}{x}=$ (a) $\log\left(\dfrac{b}{a}\right)$ (b) $\log ab$ (c) $\dfrac{a}{b}$ (d) $\log\left(\dfrac{a…Preview
- Q21Calculate: $\displaystyle\lim_{x\to 3}\dfrac{x^2-6x+5}{x^3-8x+7}$Preview
- Q22$\displaystyle\lim_{x\to 3} \lfloor x \rfloor =$ (a) Value does not exist (b) $2$ (c) $0$ (d) $3$Preview
- Q23Find the positive integer 'n' so that $\displaystyle\lim_{x\to 2} \dfrac{x^n - 2^n}{x - 2} = 12$Preview
- Q24(a) Prove that $\displaystyle\lim_{\theta\to 0} \dfrac{\sin\theta}{\theta} = 1$ **OR** (b) A factory has two Machines - I and II. Machine -…Preview
- Q25$f(x)=\dfrac{1}{x}$ is continuous at: (a) $(-\infty, 0]$ (b) $\mathbb{R}$ (c) $[0, \infty)$ (d) $\mathbb{R}-\{0\}$Preview
- Q26Prove that $f(x)=2x^2+3x-5$ is continuous at all points in $\mathbb{R}$.Preview
- Q27Calculate $\lim\limits_{x\to 1} \dfrac{x^{11}-1}{x-1}$Preview
- Q28Show that: $\displaystyle\lim_{x\to 0^+} x\left[\left\lfloor\dfrac{1}{x}\right\rfloor+\left\lfloor\dfrac{2}{x}\right\rfloor+\ldots+\left\lfl…Preview