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Mathematics · Class 11 Science

Ch 16Limits — Class 11 Mathematics, concept-first.

7.1.1 Limit of a function. Suppose x is a variable and a is a constant. If x takes values closer and closer to a but never actually equal to a, we say x tends to a, written . When x approaches from values larger than a (for example ) we write ; when it approaches from values smaller than a we write .

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Limits by Factorization

When a rational function's limit produces the indeterminate form zero over zero at x equals a, it is a certain sign that (x minus a) divides evenly into both the numerator polynomial and the denominator polynomial.

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7.1

Meaning and Algebra of Limits

7.1.1 Limit of a function. Suppose x is a variable and a is a constant. If x takes values closer and closer to a but never actually equal to a, we say x tends to a, written .

7.2

Method of Factorization

P(x) and Q(x) are polynomials in x, and . To evaluate : (1) if , the limit is simply — an ordinary substitution, no factoring needed; (2) if then divides ; if does NOT also divide , the limit does not…

7.3

Method of Rationalization

If the expression inside a limit contains a square root (or, later, a trigonometric function), it can often be simplified by multiplying numerator and denominator by the radical's rationalizing (conju…

7.4

Limit of a Trigonometric Function

Standard trigonometric substitution limits. Because and are continuous everywhere, and always hold by direct substitution — these two facts, combined with ordinary trigonometric identities, are the st…

7.5

Substitution Method

For a trigonometric limit as where a is a fixed non-zero angle (typically or ), the substitution — so and, crucially, as — converts the problem into a limit at , where the standard corollaries of sect…

7.6

Limits of Exponential and Logarithmic Functions

The chapter states, without re-proving, eight standard exponential and logarithmic limits (listed in the table note above), the two most-used being (any base ) and , together with the logarithmic coun…

7.7

Limit at Infinity and Infinite Limit

7.7.1 Limit at infinity. For , as x grows without bound (through larger and larger positive values, ) the value shrinks toward ; as x runs through larger and larger NEGATIVE values () also shrinks tow…

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137 Q
+Show 15 questions15 questions
  1. Q99Select the correct answer from the given alternatives. $\displaystyle\lim_{x\to 2}\left(\frac{x^4-16}{x^2-5x+6}\right)=$ (A) 23 (B) 32 (C) $…Free
  2. Q100$\displaystyle\lim_{x\to -2}\left(\frac{x^7+128}{x^3+8}\right)=$ (A) $\dfrac{56}{3}$ (B) $\dfrac{112}{3}$ (C) $\dfrac{121}{3}$ (D) $\dfrac{2…Free
  3. Q101$\displaystyle\lim_{x\to 3}\left(\frac{1}{x^2-11x+24}+\frac{1}{x^2-x-6}\right)=$ (A) $-\dfrac{2}{25}$ (B) $\dfrac{2}{25}$ (C) $\dfrac{7}{25}…Free
  4. Q102$\displaystyle\lim_{x\to 5}\left(\frac{\sqrt{x+4}-3}{\sqrt{3x-11}-2}\right)=$ (A) $\dfrac{-2}{9}$ (B) $\dfrac{2}{7}$ (C) $\dfrac{5}{9}$ (D)…Preview
  5. Q103$\displaystyle\lim_{x\to \pi/3}\left(\frac{\tan^2x-3}{\sec^3x-8}\right)=$ (A) $1$ (B) $\dfrac{1}{2}$ (C) $\dfrac{1}{3}$ (D) $\dfrac{1}{4}$Preview
  6. Q104$\displaystyle\lim_{x\to 0}\left(\frac{5\sin x-x\cos x}{2\tan x-3x^2}\right)=$ (A) $0$ (B) $1$ (C) $2$ (D) $3$Preview
  7. Q105$\displaystyle\lim_{x\to \pi/2}\left[\frac{3\cos x+\cos 3x}{(2x-\pi)^3}\right]=$ (A) $\dfrac{3}{2}$ (B) $\dfrac{1}{2}$ (C) $-\dfrac{1}{2}$ (…Preview
  8. Q106$\displaystyle\lim_{x\to 0}\left(\frac{15^x-3^x-5^x+1}{\sin^2x}\right)=$ (A) $\log 15$ (B) $\log 3+\log 5$ (C) $\log 3.\log 5$ (D) $3\log 5$Preview
  9. Q107$\displaystyle\lim_{x\to 0}\left(\frac{3+5x}{3-4x}\right)^{1/x}=$ (A) $e^3$ (B) $e^6$ (C) $e^9$ (D) $e^{-3}$Preview
  10. Q108$\displaystyle\lim_{x\to 0}\left[\frac{\log(5+x)-\log(5-x)}{\sin x}\right]=$ (A) $\dfrac{3}{2}$ (B) $-\dfrac{5}{2}$ (C) $-\dfrac{1}{2}$ (D)…Preview
  11. Q109$\displaystyle\lim_{x\to \pi/2}\left(\frac{3^{\cos x}-1}{\dfrac{\pi}{2}-x}\right)=$ (A) $1$ (B) $\log 3$ (C) $3^{\pi/2}$ (D) $3\log 3$Preview
  12. Q110$\displaystyle\lim_{x\to 0}\left(\frac{x\cdot\log(1+3x)}{(e^{3x}-1)^2}\right)=$ (A) $\dfrac{1}{e^9}$ (B) $\dfrac{1}{e^3}$ (C) $\dfrac{1}{9}$…Preview
  13. Q111$\displaystyle\lim_{x\to 0}\left[\frac{(3^{\sin x}-1)^3}{(3^x-1)\cdot\tan x\cdot\log(1+x)}\right]=$ (A) $3\log 3$ (B) $2\log 3$ (C) $(\log 3…Preview
  14. Q112$\displaystyle\lim_{x\to 3}\left[\frac{5^{x-3}-4^{x-3}}{\sin(x-3)}\right]=$ (A) $\log 5-4$ (B) $\log \dfrac{5}{4}$ (C) $\dfrac{\log 5}{\log…Preview
  15. Q113$\displaystyle\lim_{x\to \infty}\left[\frac{(2x+3)^7(x-5)^3}{(2x-5)^{10}}\right]=$ (A) $\dfrac{3}{8}$ (B) $\dfrac{1}{8}$ (C) $\dfrac{1}{6}$…Preview
+Show 24 questions24 questions
  1. Q114$\displaystyle\lim_{x\to 0}\left[\frac{(1-x)^5-1}{(1-x)^3-1}\right]$Free
  2. Q115$\displaystyle\lim_{x\to 0}[x]$ ($[*]$ is a greatest integer function.)Free
  3. Q116If $f(r)=\pi r^2$ then find $\displaystyle\lim_{h\to 0}\left[\frac{f(r+h)-f(r)}{h}\right]$Free
  4. Q117$\displaystyle\lim_{x\to 0}\left[\frac{x}{|x|+x^2}\right]$Preview
  5. Q118Find the limit of the function, if it exists, at $x=1$: $f(x)=\begin{cases}7-4x & \text{for } x<1\\ x^2+2 & \text{for } x\ge 1\end{cases}$Preview
  6. Q119Given that $7x\le f(x)\le 3x^2-6$ for all $x$. Determine the value of $\displaystyle\lim_{x\to 3}f(x)$Preview
  7. Q120$\displaystyle\lim_{x\to 0}\left[\frac{\sec(x^2)-1}{x^4}\right]$Preview
  8. Q121$\displaystyle\lim_{x\to 0}\left[\frac{e^x+e^{-x}-2}{x\cdot\tan x}\right]$Preview
  9. Q122$\displaystyle\lim_{x\to 0}\left[\frac{x(6^x-3^x)}{\cos(6x)-\cos(4x)}\right]$Preview
  10. Q123$\displaystyle\lim_{x\to 0}\left[\frac{a^{3x}-a^{2x}-a^{x}+1}{x\cdot\tan x}\right]$Preview
  11. Q124$\displaystyle\lim_{x\to a}\left[\frac{\sin x-\sin a}{x-a}\right]$Preview
  12. Q125$\displaystyle\lim_{x\to 2}\left[\frac{\log x-\log 2}{x-2}\right]$Preview
  13. Q126$\displaystyle\lim_{x\to 1}\left[\frac{ab^x-a^xb}{x^2-1}\right]$Preview
  14. Q127$\displaystyle\lim_{x\to 0}\left[\frac{(5^x-1)^2}{(2^x-1)\log(1+x)}\right]$Preview
  15. Q128$\displaystyle\lim_{x\to \infty}\left[\frac{(2x+1)^2(7x-3)^3}{(5x+2)^5}\right]$Preview
  16. Q129$\displaystyle\lim_{x\to a}\left[\frac{x\cos a-a\cos x}{x-a}\right]$Preview
  17. Q130$\displaystyle\lim_{x\to \pi/4}\left[\frac{(\sin x-\cos x)^2}{\sqrt2-\sin x-\cos x}\right]$Preview
  18. Q131$\displaystyle\lim_{x\to 1}\left[\frac{2^{2x-2}-2^{x}+1}{\sin^2(x-1)}\right]$Preview
  19. Q132$\displaystyle\lim_{x\to 1}\left[\frac{4^{x-1}-2^{x}+1}{(x-1)^2}\right]$Preview
  20. Q133$\displaystyle\lim_{x\to 1}\left[\frac{\sqrt{x}-1}{\log x}\right]$Preview
  21. Q134$\displaystyle\lim_{x\to 0}\left(\frac{\sqrt{1-\cos x}}{x}\right)$Preview
  22. Q135$\displaystyle\lim_{x\to 1}\left(\frac{x+3x^2+5x^3+\dots\dots+(2n-1)x^n-n^2}{x-1}\right)$Preview
  23. Q136$\displaystyle\lim_{x\to 0}\left\{\frac{1}{x^{12}}\left[1-\cos\left(\frac{x^2}{2}\right)-\cos\left(\frac{x^4}{4}\right)+\cos\left(\frac{x^2}…Preview
  24. Q137$\displaystyle\lim_{x\to \infty}\left(\frac{8x^2+5x+3}{2x^2-7x-5}\right)^{\frac{4x+3}{8x-1}}$Preview
+Show 3 questions3 questions
  1. Q1$\displaystyle\lim_{z\to -3}\left[\frac{\sqrt{z+6}}{z}\right]$Free
  2. Q2$\displaystyle\lim_{y\to -3}\left[\frac{y^5+243}{y^3+27}\right]$Preview
  3. Q3$\displaystyle\lim_{z\to -5}\left[\frac{\dfrac{1}{z}+\dfrac{1}{5}}{z+5}\right]$Preview
+Show 4 questions4 questions
  1. Q4$\displaystyle\lim_{x\to 3}\left[\frac{\sqrt{2x+6}}{x}\right]$Free
  2. Q5$\displaystyle\lim_{x\to 2}\left[\frac{x^{-3}-2^{-3}}{x-2}\right]$Free
  3. Q6$\displaystyle\lim_{x\to 5}\left[\frac{x^3-125}{x^5-3125}\right]$Preview
  4. Q7If $\displaystyle\lim_{x\to 1}\left[\frac{x^4-1}{x-1}\right]=\lim_{x\to a}\left[\frac{x^3-a^3}{x-a}\right]$, find all possible values of $a$…Preview
+Show 9 questions9 questions
  1. Q8$\displaystyle\lim_{x\to 1}\left[\frac{x+x^2+x^3+\dots+x^n-n}{x-1}\right]$Free
  2. Q9$\displaystyle\lim_{x\to 7}\left[\frac{\left(\sqrt[3]{x}-\sqrt[3]{7}\right)\left(\sqrt[3]{x}+\sqrt[3]{7}\right)}{x-7}\right]$Free
  3. Q10If $\displaystyle\lim_{x\to 5}\left[\frac{x^k-5^k}{x-5}\right]=500$, find all possible values of $k$.Free
  4. Q11$\displaystyle\lim_{x\to 0}\left[\frac{(1-x)^8-1}{(1-x)^2-1}\right]$Preview
  5. Q12$\displaystyle\lim_{x\to 0}\left[\frac{\sqrt[3]{1+x}-\sqrt{1+x}}{x}\right]$Preview
  6. Q13$\displaystyle\lim_{y\to 1}\left[\frac{2y-2}{\sqrt[3]{7+y}-2}\right]$Preview
  7. Q14$\displaystyle\lim_{z\to a}\left[\frac{(z+2)^{3/2}-(a+2)^{3/2}}{z-a}\right]$Preview
  8. Q15$\displaystyle\lim_{x\to 7}\left[\frac{x^3-343}{\sqrt{x}-\sqrt{7}}\right]$Preview
  9. Q16$\displaystyle\lim_{x\to 1}\left(\frac{x+x^3+x^5+\dots+x^{2n-1}-n}{x-1}\right)$Preview
+Show 4 questions4 questions
  1. Q17In the following example, given $\epsilon>0$, find a $\delta>0$ such that whenever $|x-a|<\delta$, we must have $|f(x)-l|<\epsilon$: $\displ…Free
  2. Q18In the following example, given $\epsilon>0$, find a $\delta>0$ such that whenever $|x-a|<\delta$, we must have $|f(x)-l|<\epsilon$: $\displ…Free
  3. Q19In the following example, given $\epsilon>0$, find a $\delta>0$ such that whenever $|x-a|<\delta$, we must have $|f(x)-l|<\epsilon$: $\displ…Preview
  4. Q20In the following example, given $\epsilon>0$, find a $\delta>0$ such that whenever $|x-a|<\delta$, we must have $|f(x)-l|<\epsilon$: $\displ…Preview
+Show 5 questions5 questions
  1. Q21$\displaystyle\lim_{z\to 2}\left[\frac{z^2-5z+6}{z^2-4}\right]$Free
  2. Q22$\displaystyle\lim_{x\to -3}\left[\frac{x+3}{x^2+4x+3}\right]$Free
  3. Q23$\displaystyle\lim_{y\to 0}\left[\frac{5y^3+8y^2}{3y^4-16y^2}\right]$Preview
  4. Q24$\displaystyle\lim_{x\to -2}\left[\frac{-2x-4}{x^3+2x^2}\right]$Preview
  5. Q25$\displaystyle\lim_{x\to 3}\left[\frac{x^2+2x-15}{x^2-5x+6}\right]$Preview
+Show 6 questions6 questions
  1. Q26$\displaystyle\lim_{u\to 1}\left[\frac{u^4-1}{u^3-1}\right]$Free
  2. Q27$\displaystyle\lim_{x\to 3}\left[\frac{1}{x-3}-\frac{9x}{x^3-27}\right]$Free
  3. Q28$\displaystyle\lim_{x\to 2}\left[\frac{x^3-4x^2+4x}{x^2-1}\right]$Preview
  4. Q29$\displaystyle\lim_{\Delta x\to 0}\left[\frac{(x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1)}{\Delta x}\right]$Preview
  5. Q30$\displaystyle\lim_{x\to \sqrt2}\left[\frac{x^2+x\sqrt2-4}{x^2-3x\sqrt2+4}\right]$Preview
  6. Q31$\displaystyle\lim_{x\to 2}\left[\frac{x^3-7x+6}{x^3-7x^2+16x-12}\right]$Preview
+Show 5 questions5 questions
  1. Q32$\displaystyle\lim_{y\to 1/2}\left[\frac{1-8y^3}{y-4y^3}\right]$Free
  2. Q33$\displaystyle\lim_{x\to 1}\left[\frac{x-2}{x^2-x}-\frac{1}{x^3-3x^2+2x}\right]$Free
  3. Q34$\displaystyle\lim_{x\to 1}\left[\frac{x^4-3x^2+2}{x^3-5x^2+3x+1}\right]$Preview
  4. Q35$\displaystyle\lim_{x\to 1}\left[\frac{x+2}{x^2-5x+4}+\frac{x-4}{3(x^2-3x+2)}\right]$Preview
  5. Q36$\displaystyle\lim_{x\to a}\left[\frac{1}{x^2-3ax+2a^2}+\frac{1}{2x^2-3ax+a^2}\right]$Preview
+Show 4 questions4 questions
  1. Q37$\displaystyle\lim_{x\to 0}\left[\frac{\sqrt{6+x+x^2}-\sqrt6}{x}\right]$Free
  2. Q38$\displaystyle\lim_{x\to 3}\left[\frac{\sqrt{2x+3}-\sqrt{4x-3}}{x^2-9}\right]$Free
  3. Q39$\displaystyle\lim_{y\to 0}\left[\frac{\sqrt{1-y^2}-\sqrt{1+y^2}}{y^2}\right]$Preview
  4. Q40$\displaystyle\lim_{x\to 2}\left[\frac{\sqrt{2+x}-\sqrt{6-x}}{\sqrt{x}-\sqrt2}\right]$Preview
+Show 5 questions5 questions
  1. Q41$\displaystyle\lim_{x\to a}\left[\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\right]$Free
  2. Q42$\displaystyle\lim_{x\to 2}\left[\frac{x^2-4}{\sqrt{x+2}-\sqrt{3x-2}}\right]$Free
  3. Q43$\displaystyle\lim_{x\to 2}\left[\frac{\sqrt{1+\sqrt{2+x}}-\sqrt3}{x-2}\right]$Preview
  4. Q44$\displaystyle\lim_{y\to 0}\left[\frac{\sqrt{a+y}-\sqrt{a}}{y\sqrt{a+y}}\right]$Preview
  5. Q45$\displaystyle\lim_{x\to 0}\left[\frac{\sqrt{x^2+9}-\sqrt{2x^2+9}}{\sqrt{3x^2+4}-\sqrt{2x^2+4}}\right]$Preview
+Show 5 questions5 questions
  1. Q46$\displaystyle\lim_{x\to 1}\left[\frac{x^2+x\sqrt{x}-2}{x-1}\right]$Free
  2. Q47$\displaystyle\lim_{x\to 0}\left[\frac{\sqrt{1+x^2}-\sqrt{1+x}}{\sqrt{1+x^3}-\sqrt{1+x}}\right]$Free
  3. Q48$\displaystyle\lim_{x\to 4}\left[\frac{x^2+x-20}{\sqrt{3x+4}-4}\right]$Preview
  4. Q49$\displaystyle\lim_{z\to 4}\left[\frac{3-\sqrt{5+z}}{1-\sqrt{5-z}}\right]$Preview
  5. Q50$\displaystyle\lim_{x\to 0}\left(\frac{3}{x\sqrt{9-x}}-\frac{1}{x}\right)$Preview
+Show 4 questions4 questions
  1. Q51$\displaystyle\lim_{\theta\to 0}\left[\frac{\sin(m\theta)}{\tan(n\theta)}\right]$Free
  2. Q52$\displaystyle\lim_{\theta\to 0}\left[\frac{1-\cos 2\theta}{\theta^2}\right]$Free
  3. Q53$\displaystyle\lim_{x\to 0}\left[\frac{x\cdot\tan x}{1-\cos x}\right]$Preview
  4. Q54$\displaystyle\lim_{x\to 0}\left(\frac{\sec x-1}{x^2}\right)$Preview
+Show 3 questions3 questions
  1. Q55$\displaystyle\lim_{x\to 0}\left[\frac{1-\cos(nx)}{1-\cos(mx)}\right]$Free
  2. Q56$\displaystyle\lim_{x\to \pi/6}\left[\frac{2-\operatorname{cosec}x}{\cot^2x-3}\right]$Preview
  3. Q57$\displaystyle\lim_{x\to \pi/4}\left[\frac{\cos x-\sin x}{\cos 2x}\right]$Preview
+Show 4 questions4 questions
  1. Q58$\displaystyle\lim_{x\to 0}\left[\frac{\cos(ax)-\cos(bx)}{\cos(cx)-1}\right]$Free
  2. Q59$\displaystyle\lim_{x\to \pi}\left[\frac{\sqrt{1-\cos x}-\sqrt2}{\sin^2x}\right]$Free
  3. Q60$\displaystyle\lim_{x\to \pi/4}\left[\frac{\tan^2x-\cot^2x}{\sec x-\operatorname{cosec}x}\right]$Preview
  4. Q61$\displaystyle\lim_{x\to \pi/6}\left[\frac{2\sin^2x+\sin x-1}{2\sin^2x-3\sin x+1}\right]$Preview
+Show 5 questions5 questions
  1. Q62$\displaystyle\lim_{x\to \pi/2}\left[\frac{\operatorname{cosec}x-1}{\left(\dfrac{\pi}{2}-x\right)^2}\right]$Free
  2. Q63$\displaystyle\lim_{x\to a}\frac{\sin x-\sin a}{\sqrt[5]{x}-\sqrt[5]{a}}$Free
  3. Q64$\displaystyle\lim_{x\to \pi}\left[\frac{\sqrt{5+\cos x}-2}{(\pi-x)^2}\right]$Preview
  4. Q65$\displaystyle\lim_{x\to \pi/6}\left[\frac{\cos x-\sqrt3\sin x}{\pi-6x}\right]$Preview
  5. Q66$\displaystyle\lim_{x\to 1}\left[\frac{1-x^2}{\sin \pi x}\right]$Preview
+Show 5 questions5 questions
  1. Q67$\displaystyle\lim_{x\to \pi/6}\left[\frac{2\sin x-1}{\pi-6x}\right]$Free
  2. Q68$\displaystyle\lim_{x\to \pi/4}\left[\frac{\sqrt2-\cos x-\sin x}{(4x-\pi)^2}\right]$Free
  3. Q69$\displaystyle\lim_{x\to \pi/6}\left[\frac{2-\sqrt3\cos x-\sin x}{(6x-\pi)^2}\right]$Preview
  4. Q70$\displaystyle\lim_{x\to a}\left[\frac{\sin\left(\sqrt{x}\right)-\sin\left(\sqrt{a}\right)}{x-a}\right]$Preview
  5. Q71$\displaystyle\lim_{x\to \pi/2}\left[\frac{\cos 3x+3\cos x}{(2x-\pi)^3}\right]$Preview
+Show 5 questions5 questions
  1. Q72$\displaystyle\lim_{x\to 0}\left[\frac{9^x-5^x}{4^x-1}\right]$Free
  2. Q73$\displaystyle\lim_{x\to 0}\left[\frac{5^x+3^x-2^x-1}{x}\right]$Free
  3. Q74$\displaystyle\lim_{x\to 0}\left(\frac{a^x+b^x+c^x-3}{\sin x}\right)$Preview
  4. Q75$\displaystyle\lim_{x\to 0}\left(\frac{6^x+5^x+4^x-3^{x+1}}{\sin x}\right)$Preview
  5. Q76$\displaystyle\lim_{x\to 0}\left(\frac{8^{\sin x}-2^{\tan x}}{e^{2x}-1}\right)$Preview
+Show 6 questions6 questions
  1. Q77$\displaystyle\lim_{x\to 0}\left[\frac{3^x+3^{-x}-2}{x\cdot\tan x}\right]$Free
  2. Q78$\displaystyle\lim_{x\to 0}\left[\frac{3+x}{3-x}\right]^{1/x}$Free
  3. Q79$\displaystyle\lim_{x\to 0}\left[\frac{5x+3}{3-2x}\right]^{2/x}$Preview
  4. Q80$\displaystyle\lim_{x\to 0}\left[\log(3-x)-\log(3+x)\right]/x$Preview
  5. Q81$\displaystyle\lim_{x\to 0}\left[\frac{4x+1}{1-4x}\right]^{1/x}$Preview
  6. Q82$\displaystyle\lim_{x\to 0}\left[\frac{5+7x}{5-3x}\right]^{1/3x}$Preview
+Show 5 questions5 questions
  1. Q83$\displaystyle\lim_{x\to 0}\left[\frac{a^x-b^x}{\sin(4x)-\sin(2x)}\right]$Free
  2. Q84$\displaystyle\lim_{x\to 0}\left[\frac{(2^x-1)^3}{(3^x-1)\cdot\sin x\cdot \log(1+x)}\right]$Free
  3. Q85$\displaystyle\lim_{x\to 0}\left[\frac{15^x-5^x-3^x+1}{x\cdot\sin x}\right]$Preview
  4. Q86$\displaystyle\lim_{x\to 0}\left[\frac{(25)^x-2(5)^x+1}{x\cdot \sin x}\right]$Preview
  5. Q87$\displaystyle\lim_{x\to 0}\left[\frac{(49)^x-2(35)^x+(25)^x}{\sin x\cdot\log(1+2x)}\right]$Preview
+Show 3 questions3 questions
  1. Q88$\displaystyle\lim_{x\to \infty}\left[\frac{ax^3+bx^2+cx+d}{ex^3+fx^2+gx+h}\right]$Free
  2. Q89$\displaystyle\lim_{x\to \infty}\left[\frac{x^3+3x+2}{(x+4)(x-6)(x-3)}\right]$Preview
  3. Q90$\displaystyle\lim_{x\to \infty}\left[\frac{7x^2+5x-3}{8x^2-2x+7}\right]$Preview
+Show 3 questions3 questions
  1. Q91$\displaystyle\lim_{x\to \infty}\left[\frac{7x^2+2x-3}{\sqrt{x^4+x+2}}\right]$Free
  2. Q92$\displaystyle\lim_{x\to \infty}\left[\sqrt{x^2+4x+16}-\sqrt{x^2+16}\right]$Preview
  3. Q93$\displaystyle\lim_{x\to \infty}\left[\sqrt{x^4+4x^2}-x^2\right]$Preview
+Show 5 questions5 questions
  1. Q94$\displaystyle\lim_{x\to \infty}\left[\frac{(3x^2+4)(4x^2-6)(5x^2+2)}{4x^6+2x^4-1}\right]$Free
  2. Q95$\displaystyle\lim_{x\to \infty}\left[\frac{(3x-4)^3(4x+3)^4}{(3x+2)^7}\right]$Free
  3. Q96$\displaystyle\lim_{x\to \infty}\left[\sqrt{x}\left(\sqrt{x+1}-\sqrt{x}\right)\right]$Preview
  4. Q97$\displaystyle\lim_{x\to \infty}\left[\frac{(2x-1)^{20}(3x-1)^{30}}{(2x+1)^{50}}\right]$Preview
  5. Q98$\displaystyle\lim_{x\to \infty}\left[\frac{\sqrt{x^2+5}-\sqrt{x^2-3}}{\sqrt{x^2+3}-\sqrt{x^2+1}}\right]$Preview