Q.Discuss the continuity of the function $f$ given by $f(x) = x^3 + x^2 - 1$.

Showing the 12 most recent of 32 on this concept.

1. AP EAPCET 2023Set eng-2023-05-18-FN1 markMCQ
   Q.Which one of the following function is discontinuous at x=1? (A) $f(x)=\sin^2x+\tan^2x+\cos^2x-\sec^2x$ (B) $f(x)=\dfrac{1}{1+2^{\sin x}}$ (C) $f(x)=\begin{cases}\dfrac{x-1}{|x-1|+2(x-1)^2}, & x\neq1\\ 1, & x=1\end{cases}$ (D) $f(x)=e^{x}+5$

   ›Reveal solutionSolution

   Options (A), (B), (D) simplify to functions that are continuous everywhere; option (C)'s one-sided limits at $x=1$ disagree ($+1$ from the right, $-1$ from the left), so it alone is discontinuous at $x=1$.

   Concept and Intuition

   For piecewise/rational-looking expressions, check whether the identity $\sin^2x+\cos^2x=1$ and $\sec^2x-\tan^2x=1$ collapse them to constants (continuous), whether a denominator can vanish, and — for absolute-value expressions — whether the left and right limits genuinely agree.

   Step-by-Step Solution

   1. (A) $f(x)=\sin^2x+\tan^2x+\cos^2x-\sec^2x=(\sin^2x+\cos^2x)-(\sec^2x-\tan^2x)=1-1=0$ for all $x$ where $\tan,\sec$ are defined; at $x=1$ (radian) $\cos1\ne0$, so it's fine and continuous ($\equiv0$ near $x=1$).
   2. (B) $f(x)=\dfrac{1}{1+2^{\sin x}}$: since $2^{\sin x}>0$ always, the denominator never vanishes ($\ge1+2^{-1}=1.5$); continuous everywhere, including $x=1$.
   3. (C) For $x\ne1$, let $t=x-1$: $f=\dfrac{t}{|t|+2t^2}$.
      • As $t\to0^+$ ($x\to1^+$): $|t|=t$, so $f=\dfrac{t}{t+2t^2}=\dfrac{1}{1+2t}\to1$.
      • As $t\to0^-$ ($x\to1^-$): $|t|=-t$, so $f=\dfrac{t}{-t+2t^2}=\dfrac{1}{-1+2t}\to-1$. …

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2. AP EAPCET 2025Set eng-2025-05-21-FN1 markMCQ
   Q.If a real valued function $f(x)=\begin{cases}\dfrac{x^2+(a+3)x+(a+1)}{x+3}, & x\neq -3\\ -\dfrac{5}{2}, & x=-3\end{cases}$ is continuous at $x=-3$, then $\displaystyle\lim_{x\to a}(x^2+x+1) =$ (A) $\dfrac{7}{4}$ (B) $\dfrac{5}{2}$ (C) $\dfrac{4}{7}$ (D) $\dfrac{2}{5}$

   ›Reveal solutionSolution

   This tests continuity of a rational function with a removable-type discontinuity: matching the limit to the given value pins down the parameter $a$, then the asked limit is a trivial polynomial evaluation. Answer: $\dfrac{7}{4}$.

   Concept and Intuition

   A piecewise function is continuous at a point if the limit of the "elsewhere" formula, as $x$ approaches that point, equals the value assigned at that point. Here the formula is a rational function whose denominator vanishes at $x=-3$; for the limit to exist (and be finite, matching $-\tfrac52$), the numerator must also vanish there, so that $(x+3)$ cancels — this is the standard "$\frac00$ removable singularity" idea.

   Step-by-Step Solution

   1. Continuity at $x=-3$ requires $\displaystyle\lim_{x\to -3}\frac{x^2+(a+3)x+(a+1)}{x+3} = -\frac52$.
   2. Since the denominator $\to 0$, the numerator must vanish at $x=-3$ (else the limit is $\pm\infty$, not finite):

   $(-3)^2+(a+3)(-3)+(a+1) = 9-3a-9+a+1 = 1-2a = 0 \implies a=\frac12.$

   3. Check: with $a=\tfrac12$, numerator $=x^2+\tfrac72x+\tfrac32$. Factor out $(x+3)$: $x^2+\tfrac72x+\tfrac32 = (x+3)\left(x+\tfrac12\right)$ (verify: $(x+3)(x+\tfrac12)=x^2+\tfrac12x+3x+\tfrac32=x^2+\tfrac72x+\tfrac32$ ✓).
   4. So $\displaystyle\lim_{x\to -3}\frac{(x+3)(x+\frac12)}{x+3} = -3+\frac12 = -\frac52$, matching the given value — confirming $a=\tfrac12$. …

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3. AP EAPCET 2024Set eng-2024-05-22-FN1 markMCQ
   Q.Let $f(x) = \begin{cases} 0, & x = 0 \\ 2 - x, & \text{for } 0 < x < 1 \\ 2, & \text{for } x = 1 \\ \dfrac{1}{2} - x, & \text{for } 1 < x < 2 \\ \dfrac{-3}{2}, & \text{for } x \ge 2 \end{cases}$ then which of the following is true (A) $f$ is right continuous at $x = 0$ (B) $f$ is left continuous at $x = 1$ (C) $f$ is right continuous at $x = 1$ (D) $f$ is continuous at $x = 2$

   ›Reveal solutionSolution

   Compute the left-hand limit, right-hand limit, and function value at each candidate point and compare; only at $x=2$ do all three coincide.

   Concept and Intuition

   A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value there all agree. "Right continuous" only needs the right-hand limit to equal the function value; "left continuous" only needs the left-hand limit to match.

   Step-by-Step Solution

   1. At $x=0$: $f(0)=0$. Right-hand limit: $\lim_{x\to0^+}(2-x)=2$. Since $2\ne0$, $f$ is not right continuous at $0$ — (A) is false.
   2. At $x=1$: $f(1)=2$. Left-hand limit: $\lim_{x\to1^-}(2-x)=1$. Since $1\ne2$, not left continuous — (B) is false. Right-hand limit: $\lim_{x\to1^+}\left(\dfrac12-x\right)=-\dfrac12$. Since $-\dfrac12\ne2$, not right continuous either — (C) is false. …

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4. AP EAPCET 2026Set eng-2026-05-15-FN1 markMCQ
   Q.If $f(x) = \dfrac{\cos^4 x - 1}{x^2}$, $x \ne 0$ and $f(0) = 2$ is a real valued function, then (A) $\displaystyle\lim_{x \to 0} f(x)$ does not exist (B) $\displaystyle\lim_{x \to 0} f(x) = 1$ (C) f is not continuous at x = 0 (D) f is continuous at x = 0

   ›Reveal solutionSolution

   This tests factoring $\cos^4x-1$ and using the standard limit $\frac{\sin x}{x}\to1$; the limit exists and equals $-2$, but $f(0)=2$, so $f$ fails continuity at $0$.

   Concept and Intuition

   $\cos^4x-1$ factors as a difference of squares twice: $\cos^4x-1=(\cos^2x-1)(\cos^2x+1)=-\sin^2x(\cos^2x+1)$. This lets us isolate the familiar $\left(\frac{\sin x}{x}\right)^2\to1$ building block.

   Step-by-Step Solution

   1. $\cos^4x - 1 = (\cos^2x-1)(\cos^2x+1) = -\sin^2x\,(\cos^2x+1)$.
   2. $f(x) = \dfrac{-\sin^2x(\cos^2x+1)}{x^2} = -\left(\dfrac{\sin x}{x}\right)^2(\cos^2x+1)$.
   3. As $x\to0$: $\left(\dfrac{\sin x}{x}\right)^2\to1$ and $\cos^2x+1\to2$.
   4. So $\displaystyle\lim_{x\to0}f(x) = -1\times2 = -2$ — this limit exists (rules out (A)) and is not $1$ (rules out (B)). …

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5. AP EAPCET 2023Set eng-2023-05-18-AN1 markMCQ
   Q.Let $S_n=1+3x+9x^2+27x^3+\ldots n$ terms and $-\frac{1}{3}<x<\frac{1}{3}$. If $\lim_{n\to\infty}S_n=f(x)$, then $f(x)$ is discontinuous at the point $x=$ (A) $0$ (B) $\frac{1}{3}$ (C) $1$ (D) $-1$

   ›Reveal solutionSolution

   The infinite geometric series sums to $f(x)=1/(1-3x)$ on $(-1/3,1/3)$, and this function has an infinite discontinuity exactly at $x=1/3$, the edge of the interval where the series stops converging.

   Concept and Intuition

   An infinite geometric series $1+r+r^2+\dots$ converges to $\frac{1}{1-r}$ only when $|r|<1$; as $r\to1$ this sum diverges to infinity. Here $r=3x$, so the series converges for $|x|<1/3$ and the resulting closed-form function has a vertical asymptote right at the boundary $x=1/3$.

   Step-by-Step Solution

   1. $S_n = 1+3x+(3x)^2+\dots+(3x)^{n-1}$, a geometric series with first term $1$ and common ratio $3x$.
   2. Sum: $S_n=\dfrac{1-(3x)^n}{1-3x}$ (for $3x\ne1$).
   3. For $-\tfrac13<x<\tfrac13$, $|3x|<1$, so $(3x)^n\to0$ as $n\to\infty$.
   4. Hence $f(x)=\lim_{n\to\infty}S_n = \dfrac{1}{1-3x}$.
   5. This function $f(x)=\frac{1}{1-3x}$ is undefined/blows up exactly where $1-3x=0$, i.e. $x=\tfrac13$ — an infinite discontinuity right at the edge of the domain of convergence. …

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6. AP EAPCET 2023Set eng-2023-05-15-FN1 markMCQ
   Q.If $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\begin{cases} \dfrac{\sin x - \sin\dfrac{x}{2}}{x}, & x<0 \\[2mm] \dfrac{\sqrt{x^2+x}-\sqrt{x}}{x^{3/2}}, & x>0 \end{cases}$ is continuous on $\mathbb{R}$, then $f(0)=$ (A) $1/2$ (B) $3/2$ (C) $1$ (D) $-1$

   ›Reveal solutionSolution

   This tests continuity of a piecewise function at the junction point using standard small-$x$ approximations; both one-sided limits equal $1/2$.

   Concept and Intuition

   A piecewise function is continuous at a boundary point only if the left-hand limit, right-hand limit, and the defined value at that point all agree. Here $f(0)$ isn't given directly by either branch (both blow up as $0/0$), so we must compute the limits from each side and set $f(0)$ equal to their common value.

   Step-by-Step Solution

   1. Left-hand limit ($x\to 0^-$):

   $\lim_{x\to0^-}\frac{\sin x-\sin(x/2)}{x}$

   Use the small-angle expansion $\sin t = t - \dfrac{t^3}{6}+\cdots$:

   $\sin x - \sin\frac{x}{2} = \left(x-\frac{x^3}{6}\right)-\left(\frac{x}{2}-\frac{x^3}{48}\right)+\cdots = \frac{x}{2} + O(x^3)$

   Dividing by $x$: the limit is $\dfrac12$.

   2. Right-hand limit ($x\to 0^+$):

   $\lim_{x\to0^+}\frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3/2}} = \lim_{x\to0^+}\frac{\sqrt{x}\left(\sqrt{x+1}-1\right)}{x^{3/2}} = \lim_{x\to0^+}\frac{\sqrt{x+1}-1}{x}$

   Using $\sqrt{1+x}\approx 1+\dfrac{x}{2}-\dfrac{x^2}{8}+\cdots$:

   $\frac{\sqrt{x+1}-1}{x}\approx \frac{x/2}{x} = \frac12$ …

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7. AP EAPCET 2021Set eng-2021-08-20-AN1 markMCQ
   Q.If $f(x) = \dfrac{\log_e\left(1 + x^2(\tan x)\right)}{\sin x^3}$, $x \neq 0$ is to be continuous at $x = 0$, then $f(0)$ must be equal to ______ (A) $1$ (B) $0$ (C) $\dfrac{1}{2}$ (D) $-1$

   ›Reveal solutionSolution

   Using the small-angle equivalences $\log_e(1+u)\sim u$, $\tan x\sim x$, and $\sin x^3\sim x^3$ near $x=0$, the limit of $f(x)$ works out to $1$, so $f(0)=1$ is required for continuity.

   Concept and Intuition

   For a function to be continuous at a removable-looking point like $x=0$, its value there must be defined equal to the limit of the function as $x\to0$. Standard small-$x$ equivalences ($\log_e(1+u)\sim u$ for small $u$; $\tan x\sim x$; $\sin x\sim x$) let us evaluate the $\frac{0}{0}$-type limit cleanly.

   Step-by-Step Solution

   1. As $x\to0$, let $u=x^2\tan x$. Since $\tan x\sim x$ for small $x$, $u\sim x^2\cdot x=x^3\to0$.
   2. Using $\log_e(1+u)\sim u$ for small $u$: $\log_e(1+x^2\tan x)\sim x^2\tan x\sim x^3$.
   3. For the denominator, $\sin(x^3)\sim x^3$ as $x\to0$ (since $\sin\theta\sim\theta$ for small $\theta$, here $\theta=x^3\to0$).
   4. So $f(x)=\dfrac{\log_e(1+x^2\tan x)}{\sin(x^3)}\to\dfrac{x^3}{x^3}=1$ as $x\to0$. …

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8. AP EAPCET 2024Set eng-2024-05-19-AN1 markMCQ
   Q.The function $f(x)=\begin{cases}\dfrac{2}{5-x}, & x<3\\ 5-x, & x \ge 3\end{cases}$ is (A) left discontinuous at $x=3$ (B) left continuous at $x=3$ (C) right discontinuous at $x=5$ (D) discontinuous at $x=5$

   ›Reveal solutionSolution

   Checking the one-sided limits at the junction $x=3$ shows the function jumps from $1$ to $2$ on approach from the left, i.e. it is left-discontinuous (but right-continuous) at $x=3$; there is nothing special happening at $x=5$.

   Concept and Intuition

   A piecewise function can fail to be continuous from only one side at a junction point if the junction point itself is included in one branch. Here $x=3$ is the boundary, and it belongs to the $5-x$ branch (since that branch is defined for $x\ge3$), so we must check whether the other branch's limit (from the left) agrees with the actual function value.

   Step-by-Step Solution

   1. Function value at $x=3$: since $x\ge3$ uses $f(x)=5-x$, $f(3)=5-3=2$.
   2. Left-hand limit as $x\to3^-$: uses $f(x)=\dfrac{2}{5-x}$, so $\lim_{x\to3^-}\dfrac{2}{5-x}=\dfrac{2}{5-3}=1$.
   3. Since LHL $=1\ne f(3)=2$, the function is discontinuous when approached from the left — i.e. left discontinuous at $x=3$.
   4. Right-hand limit as $x\to3^+$: still $5-x\to2=f(3)$, so it is right-continuous at $x=3$ — this rules out "left continuous" (option B, false). …

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9. AP EAPCET 2024Set eng-2024-05-18-FN1 markMCQ
   Q.$f(x)=\begin{cases}\dfrac{(2x^2-ax+1)-(ax^2+3bx+2)}{x+1}, & \text{if } x\neq -1\\ k, & \text{if } x=-1\end{cases}$ is a real valued function. If $a,b,k\in R$ and $f$ is continuous on $R$ then $k=$ (A) $-\dfrac{1}{3}$ (B) 6 (C) $a-2$ (D) $a-3$

   ›Reveal solutionSolution

   Continuity at the removable-type point requires the numerator to vanish there (fixing $b$) and then simplifying the quotient to evaluate $k$ as a limit. The answer is $k=a-3$.

   Concept and Intuition

   A rational expression $\dfrac{N(x)}{x+1}$ can only have a finite limit as $x\to-1$ if $N(-1)=0$ (otherwise the limit is $\pm\infty$ and no choice of $k$ can make $f$ continuous there). So the first job is to use that vanishing condition to pin down any free constant, then simplify by cancelling the common factor.

   Step-by-Step Solution

   1. Numerator: $(2x^2-ax+1)-(ax^2+3bx+2)=(2-a)x^2-(a+3b)x-1$.
   2. At $x=-1$: $(2-a)(1)+(a+3b)-1=1+3b$. For the limit (hence continuity) to exist finitely, this must be $0$: $b=-\dfrac13$.
   3. With $b=-1/3$, numerator becomes $(2-a)x^2-(a-1)x-1$.
   4. Factor out $(x+1)$: writing $(2-a)x^2-(a-1)x-1=(x+1)\left[(2-a)x-1\right]$ (verified by expansion, matching all coefficients). …

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10. AP EAPCET 2024Set eng-2024-05-21-AN1 markMCQ
    Q.If the function $f(x) = \dfrac{\sqrt{1+x}-1}{x}$ is continuous at $x = 0$ then $f(0) =$ (A) $-\dfrac{1}{2}$ (B) $\dfrac{1}{3}$ (C) $\dfrac{1}{2}$ (D) $-\dfrac{1}{3}$

    ›Reveal solutionSolution

    Rationalizing the numerator turns the $0/0$ form into a simple limit, which must equal $f(0)$ for continuity.

    Concept and Intuition

    For $f$ to be continuous at $x=0$, we need $f(0)=\displaystyle\lim_{x\to0}f(x)$. Since the given formula for $f(x)$ is $0/0$ at $x=0$, we must simplify it algebraically first.

    Step-by-Step Solution

    1. $\dfrac{\sqrt{1+x}-1}{x} = \dfrac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{x(\sqrt{1+x}+1)} = \dfrac{(1+x)-1}{x(\sqrt{1+x}+1)} = \dfrac{x}{x(\sqrt{1+x}+1)} = \dfrac{1}{\sqrt{1+x}+1}$.
    2. As $x\to0$: $\dfrac{1}{\sqrt{1}+1} = \dfrac12$. …

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11. AP EAPCET 2023Set eng-2023-05-16-AN1 markMCQ
    Q.If a function defined by $f(x) = \dfrac{(3^x - 1)^2}{\sin x \log(1+x)}$, $x \ne 0$, is continuous at $x = 0$, then $f(0) =$ (A) $2\log 3$ (B) $\log 3^2$ (C) $2 + \log 3$ (D) $(\log 3)^2$

    ›Reveal solutionSolution

    Standard small-$x$ equivalents give $f(x)\to(\ln3)^2$ as $x\to0$, so $f(0)=(\log3)^2$.

    Concept and Intuition

    For continuity at $x=0$, $f(0)$ must equal $\lim_{x\to0}f(x)$. Use the standard limits $\lim_{x\to0}\dfrac{a^x-1}{x}=\ln a$, $\lim_{x\to0}\dfrac{\sin x}{x}=1$, $\lim_{x\to0}\dfrac{\log(1+x)}{x}=1$.

    Step-by-Step Solution

    1. $(3^x-1)^2 = x^2\left(\dfrac{3^x-1}{x}\right)^2 \to x^2(\ln3)^2$ as $x\to0$.
    2. $\sin x \log(1+x) = x\cdot\dfrac{\sin x}{x}\cdot x\cdot\dfrac{\log(1+x)}{x} = x^2\cdot\dfrac{\sin x}{x}\cdot\dfrac{\log(1+x)}{x} \to x^2$ as $x\to0$.
    3. So $f(x)=\dfrac{(3^x-1)^2}{\sin x\log(1+x)} \to \dfrac{x^2(\ln3)^2}{x^2}=(\ln3)^2$. …

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12. AP EAPCET 2023Set eng-2023-05-15-AN1 markMCQ
    Q.If $f(x) = \begin{cases} \dfrac{x-[x]}{x-2}, & x>2 \\ b, & x=2 \\ \dfrac{|x^2-x-2|}{a(2+x-x^2)}, & -1<x\le 2 \\ 2a-b, & x\le -1 \end{cases}$ is continuous on $\mathbb{R}$, then $\displaystyle\lim_{x\to 0}\dfrac{\sin^2 ax + x\tan bx}{x^2} =$ (A) 0 (B) 1 (C) 2 (D) 3

    ›Reveal solutionSolution

    Continuity of the piecewise function pins down $a=1,b=1$; substituting these into the limit expression and using standard small-angle limits gives $2$.

    Concept and Intuition

    For a piecewise function to be continuous at a junction point, the one-sided limits and the defined value there must all agree. Here the junctions at $x=2$ and $x=-1$ give the equations needed to solve for the unknown constants $a,b$ before the actual limit can be evaluated.

    Step-by-Step Solution

    1. Right limit at $x=2$: for $x$ slightly $>2$, $[x]=2$, so $f(x)=\dfrac{x-2}{x-2}=1$. So $\lim_{x\to2^+}f(x)=1$.
    2. Left limit at $x=2$ (third piece): $x^2-x-2=(x-2)(x+1)$ and $2+x-x^2=-(x-2)(x+1)$. For $x$ near $2^-$: $(x-2)<0,(x+1)>0$, so $|x^2-x-2|=(2-x)(x+1)$ and $2+x-x^2=(2-x)(x+1)$ too. So the ratio simplifies to $\dfrac1a$ throughout $(-1,2)$.
    3. Continuity at $x=2$: $1=b=\dfrac1a \Rightarrow a=1,\ b=1$.
    4. Check at $x=-1$: piece 4 value $=2a-b=2-1=1$; piece 3's limit as $x\to-1^+$ is also $\dfrac1a=1$. Consistent ✓. …

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