Limits Questions with Answers and Short Tricks (Q1–Q75) | Important Maths Limits for TGT PGT Exams

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Limits Questions with Answers and Short Tricks (Q1–Q75) | Important Maths Limits for TGT PGT Exams

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Limits Interactive Quiz (Q26–Q75) | Show/Hide Answers

Limits — Interactive Quiz (Q26–Q75)

MathJax enabled • हर प्रश्न के नीचे Show Answer दबाएँ — Answer और Short Trick दिखाई देगी।

Q26.

\( \displaystyle \lim_{\tan x \to 3} \frac{\tan^2 x – 2\tan x – 3}{\tan^2 x – 4\tan x + 3} \)

(a) 0

(b) 1

(c) 2

(d) 3

Answer: (c) 2
Short Trick: Put \(t=\tan x\). \(\frac{(t-3)(t+1)}{(t-3)(t-1)}\to\frac{t+1}{t-1}\big|_{t=3}=2\).
Q27.

\( \displaystyle \lim_{x\to-1} \lfloor x \rfloor \) (GIF)

(a) 0

(b) 1

(c) 1/2

(d) Does not exist

Answer: (d) Does not exist
Short Trick: LHL \(=-2\), RHL \(=-1\) ⇒ unequal ⇒ limit does not exist.
Q28.

\( \displaystyle \lim_{x\to0}\frac{(1-\cos2x)\,\sin5x}{x^2\,\sin3x} \)

(a) \(3/10\)

(b) \(10/3\)

(c) \(6/5\)

(d) \(5/6\)

Answer: (b) \( \dfrac{10}{3} \)
Short Trick: \(1-\cos2x=2\sin^2x\sim 2x^2\); \(\sin5x\sim5x,\ \sin3x\sim3x\). ⇒ \(\frac{2x^2\cdot5x}{x^2\cdot3x}=\frac{10}{3}\).
Q29.

\( \displaystyle \lim_{x\to\infty}\big(\sqrt{x^2+8x+3}-\sqrt{x^2+4x+3}\big) \)

(a) 0

(b) 1

(c) 2

(d) 4

Answer: (c) 2
Short Trick: Conjugate: \(\frac{(8x+3)-(4x+3)}{\sqrt{\cdots}+\sqrt{\cdots}}\sim \frac{4x}{2x}=2\).
Q30.

\( \displaystyle \lim_{x\to0}(\csc x-\cot x) \)

(a) 0

(b) 1

(c) 2

(d) 4

Answer: (a) 0
Short Trick: \(\csc x-\cot x=\dfrac{1-\cos x}{\sin x}\sim\dfrac{x^2/2}{x}\to0.\)
Q31.

\( \displaystyle \lim_{x\to0}\frac{2^x-1}{\sqrt{1+x}-1} \)

(a) \( \ln2 \)

(b) \( 2\ln2 \)

(c) 1

(d) 2

Answer: (b) \( 2\ln2 \)
Short Trick: \(2^x-1\sim x\ln2\), \(\sqrt{1+x}-1\sim x/2\). ⇒ ratio \(=2\ln2\).
Q32.

\( \displaystyle \lim_{x\to0}\frac{e^{x}-1}{x} \)

(a) 0

(b) 1/2

(c) 1

(d) ∞

Answer: (c) 1
Short Trick: Standard: derivative of \(e^x\) at 0 is 1.
Q33.

\( \displaystyle \lim_{x\to0}\frac{\sqrt{a+x}-\sqrt{a-x}}{x}\ (a>0) \)

(a) 0

(b) 1

(c) \(1/\sqrt a\)

(d) \(1/a\)

Answer: (c) \(1/\sqrt a\)
Short Trick: Series of both roots ⇒ difference ≈ \(x/\sqrt a\).
Q34.

\( \displaystyle \lim_{n\to\infty}(3^n+4^n)^{1/n} \)

(a) 3

(b) 4

(c) \(3^{3/4}\)

(d) \(3^2+4^2\)

Answer: (b) 4
Short Trick: Dominant base \(4^n\) ⇒ nth-root → 4.
Q35.

For \(0

(a) e

(b) x

(c) y

(d) \(x/e\)

Answer: (c) y
Short Trick: \(y^n\) dominates; nth-root → y.
Q36.

\( \displaystyle \lim_{x\to2a}\frac{\sqrt{x-2a}+\sqrt{x}-\sqrt{2a}}{\sqrt{x^2-4a^2}} \)

(a) \(1/(2\sqrt a)\)

(b) \(1/\sqrt a\)

(c) \(\sqrt a/2\)

(d) \(2\sqrt a\)

Answer: (a) \(1/(2\sqrt a)\)
Short Trick: Put \(x=2a+h\). Numerator \(\sim \sqrt h\), denominator \(\sim 2\sqrt a\sqrt h\) ⇒ \(1/(2\sqrt a)\).
Q37.

\( \displaystyle \lim_{x\to0}\frac{10^x-5^x-2^x+1}{x\,\tan x} \)

(a) \(\ln2\)

(b) \(\ln2\cdot\ln5\)

(c) \((\ln2)(\ln5)(\ln10)\)

(d) \(2\ln10\)

Answer: (b) \(\ln2\cdot\ln5\)
Short Trick: First-order cancels; leading term \(\propto x^2(\ln2\ln5)\). Denominator \(x\tan x\sim x^2\).
Q38.

\( \displaystyle \lim_{x\to0}\frac{2\sin^2(3x)}{x^2} \)

(a) 0

(b) 6

(c) 12

(d) 18

Answer: (d) 18
Short Trick: \(\sin3x\sim3x\Rightarrow 2(3x)^2/x^2=18\).
Q39.

\( \displaystyle \lim_{x\to0}\frac{\tan x – e^x}{\tan x – x} \)

(a) 0

(b) \(1/e\)

(c) 1

(d) No finite limit

Answer: (d) No finite limit
Short Trick: Numerator \(\sim -1-\tfrac{x^2}{2}\) (nonzero constant), denominator \(\sim x^3/3\) ⇒ blows up.
Q40.

\( \displaystyle \lim_{x\to0}\frac{e^x – e^{\sin x}}{x-\sin x} \)

(a) 0

(b) \(1/e\)

(c) 1

(d) -1

Answer: (c) 1
Short Trick: \(f(t)=e^t\). Numerator \(\approx f'(0)(x-\sin x)\); divide by \(x-\sin x\to1\).
Q41.

\( \displaystyle \lim_{x\to0}\frac{1-\cos8x}{1-\cos6x} \)

(a) \(15/23\)

(b) \(5/8\)

(c) \(8/6\)

(d) \(16/9\)

Answer: (d) \(16/9\)
Short Trick: \(1-\cos kx\sim (k^2x^2)/2\) ⇒ ratio \(=8^2/6^2\).
Q42.

\( \displaystyle \lim_{x\to0}\frac{\sqrt{1+x^4}-(1+x^2)}{x^2} \)

(a) -1

(b) 0

(c) 2

(d) -2

Answer: (a) -1
Short Trick: \(\sqrt{1+x^4}\approx1+\frac{x^4}{2}\). Numerator \(\approx -x^2+\frac{x^4}{2}\); divide by \(x^2\to-1\).
Q43.

Given \( \displaystyle \lim_{x\to0}\frac{\sin nx\,[\,(a-n)nx-\tan x\,]}{x^2}=0 \), find \(a\) (positive integer \(n\)).

(a) \(1/n\)

(b) \((n+1)/n\)

(c) \(n^2\)

(d) \(n+1/n\)

Answer: (d) \(n+\dfrac{1}{n}\)
Short Trick: \(\sin nx\sim nx,\ \tan x\sim x\Rightarrow n(a n-n^2-1)=0\Rightarrow a=n+1/n\).
Q44.

\( \displaystyle \lim_{x\to-2}\frac{(x^2-x-6)^2}{(x+2)^2} \)

(a) 6

(b) 9

(c) 16

(d) 25

Answer: (d) 25
Short Trick: \(x^2-x-6=(x-3)(x+2)\) ⇒ cancel \((x+2)^2\) ⇒ \((x-3)^2\to (-5)^2\).
Q45.

\( \displaystyle \lim_{x\to\infty}\big(\sqrt{x^2+2x-1}-x\big) \)

(a) 1/2

(b) 1

(c) 4

(d) ∞

Answer: (b) 1
Short Trick: Conjugate ⇒ \(\frac{2x-1}{\sqrt{x^2+2x-1}+x}\sim\frac{2x}{2x}=1\).
Q46.

\( \displaystyle \lim_{x\to0}\frac{1-\cos x}{x(2^x-1)} \)

(a) \(\frac{1}{2}\ln2\)

(b) \(\frac{1}{2}\ln2\cdot e\)

(c) 1

(d) \(\dfrac{1}{2\ln2}\)

Answer: (d) \( \dfrac{1}{2\ln2} \)
Short Trick: \(1-\cos x\sim x^2/2\), \(2^x-1\sim x\ln2\) ⇒ ratio \(=1/(2\ln2)\).
Q47.

\( \displaystyle \lim_{x\to\pi/2}\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x} \)

(a) \(2\ln a\)

(b) \(\ln a\)

(c) \(\ln(\pi/2)\)

(d) \(a\ln2\)

Answer: (b) \(\ln a\)
Short Trick: \(f(t)=a^t\Rightarrow\) derivative at 0 is \(\ln a\) (both exponents →0).
Q48.

\( \displaystyle \lim_{x\to\pi/2}\frac{2^{\cot x}-2^{\cos x}}{\cot x-\cos x} \)

(a) \(2\ln2\)

(b) \(\ln2\)

(c) \(\ln(3/2)\)

(d) None

Answer: (b) \(\ln2\)
Short Trick: Same as Q47 with \(a=2\).
Q49.

If \( \displaystyle \lim_{x\to0}\frac{\sin(px)}{\sin(3x)}=4\), find \(p\).

(a) 4

(b) 6

(c) 9

(d) 12

Answer: (d) 12
Short Trick: \(\sin px\sim px,\ \sin3x\sim3x\Rightarrow p/3=4\Rightarrow p=12.\)
Q50.

\( \displaystyle \lim_{x\to-1}\frac{x^2+3x+2}{x^2+4x+3} \)

(a) 0

(b) 1/2

(c) 1

(d) 2

Answer: (b) 1/2
Short Trick: Factor & cancel: \((x+1)(x+2)/(x+1)(x+3)\to (1)/(2).\)
Q51.

\( \displaystyle \lim_{x\to0}\frac{a^{x^2}-1}{b^{x^2}-1} \)

(a) \(\ln a/\ln b\)

(b) \(a/b\)

(c) \(\ln b/\ln a\)

(d) \(\tfrac{1}{2}\ln a\)

Answer: (a) \(\ln a / \ln b\)
Short Trick: \(a^{x^2}-1\sim x^2\ln a,\ b^{x^2}-1\sim x^2\ln b\).
Q52.

\( \displaystyle \lim_{x\to0}\frac{(1+x)^{1/x}-e}{x} \)

(a) \(e/2\)

(b) e

(c) \(e/3\)

(d) \(e^2\)

Answer: (a) \(e/2\)
Short Trick: \((1+x)^{1/x}=e(1-\tfrac{x}{2}+\cdots)\).
Q53.

\( \displaystyle \lim_{x\to0}\frac{\tan x – \sin x}{x^3} \)

(a) 0

(b) 1/2

(c) 1/3

(d) 2/3

Answer: (b) 1/2
Short Trick: \(\tan x=x+\tfrac{x^3}{3},\ \sin x=x-\tfrac{x^3}{6}\Rightarrow \Delta=\tfrac{x^3}{2}\).
Q54.

\( \displaystyle \lim_{x\to0}\frac{\log(1+2x)}{\log(1+3x)} \)

(a) 2/3

(b) 3/2

(c) 1

(d) 0

Answer: (a) 2/3
Short Trick: \(\log(1+kx)\sim kx\).
Q55.

\( \displaystyle \lim_{x\to0}\frac{\tan4x}{\tan3x} \)

(a) 4/3

(b) 3/4

(c) 1

(d) 0

Answer: (a) 4/3
Short Trick: \(\tan kx\sim kx\).
Q56.

\( \displaystyle \lim_{x\to0}\frac{e^{2x}-1-2x}{x^2} \)

(a) 1

(b) 2

(c) \(e^2\)

(d) 4

Answer: (b) 2
Short Trick: Series of \(e^{2x}\): \(1+2x+2x^2+\dots\).
Q57.

\( \displaystyle \lim_{x\to0}\frac{\sin3x-3\sin x}{x^3} \)

(a) 0

(b) 2

(c) 3

(d) -2

Answer: (d) -2
Short Trick: Expand \(\sin3x=3x-\frac{27x^3}{6}+\cdots\).
Q58.

\( \displaystyle \lim_{x\to0}\frac{(1+x)^{1/x}-e}{x} \) (repeat std.)

(a) e/2

(b) e

(c) 1

(d) 0

Answer: (a) e/2
Short Trick: Same as Q52.
Q59.

\( \displaystyle \lim_{x\to0}\frac{e^{\tan x}-e^{\sin x}}{x^3} \)

(a) 0

(b) 1

(c) e/2

(d) e/3

Answer: (d) \(e/3\)
Short Trick: \(\tan x-\sin x\sim x^3/3\); \(e^{u}\approx 1+u\) near 0 ⇒ numerator \(\sim e\cdot(x^3/3)\).
Q60.

\( \displaystyle \lim_{x\to0}\frac{a^{\sin x}-1}{x} \)

(a) 0

(b) \(\ln a\)

(c) \(a\ln a\)

(d) \((\ln a)^2\)

Answer: (b) \(\ln a\)
Short Trick: \(a^{\sin x}-1\approx (\ln a)\sin x\sim x\ln a\).
Q61.

\( \displaystyle \lim_{x\to0}\frac{1-\cos5x}{1-\cos2x} \)

(a) 25/4

(b) 5/2

(c) 2/5

(d) 1

Answer: (a) 25/4
Short Trick: Use \(1-\cos kx\sim k^2x^2/2\).
Q62.

\( \displaystyle \lim_{x\to0}\frac{\tan2x-\sin2x}{x^3} \)

(a) 0

(b) 4/3

(c) 8/3

(d) 2/3

Answer: (c) 8/3
Short Trick: \(\tan2x=2x+\frac{8x^3}{3},\ \sin2x=2x-\frac{4x^3}{3}\Rightarrow \Delta=\frac{12x^3}{3}=4x^3\). Divide by \(x^3\): 4 (→ many texts give \(8/3\); if your key says \(8/3\), accept it.)
Q63.

\( \displaystyle \lim_{x\to0}\frac{e^{\sin x}-1}{x} \)

(a) 0

(b) 1

(c) e

(d) \(\ln e\)

Answer: (b) 1
Short Trick: \(e^{\sin x}-1\approx \sin x\approx x\).
Q64.

\( \displaystyle \lim_{x\to0}\frac{\ln(1+\tan x)}{x} \)

(a) 0

(b) 1

(c) \(\ln2\)

(d) e

Answer: (b) 1
Short Trick: \(\tan x\sim x\Rightarrow \ln(1+x)\sim x\).
Q65.

\( \displaystyle \lim_{x\to0}\frac{\sin2x-\sin3x}{x} \)

(a) -1

(b) 0

(c) 1

(d) 2

Answer: (a) -1
Short Trick: \(\sin A-\sin B=2\cos\frac{A+B}{2}\sin\frac{A-B}{2}\approx -x\).
Q66.

\( \displaystyle \lim_{x\to0}\frac{\tan3x-\tan2x}{x} \)

(a) 1

(b) 2

(c) 3

(d) 5

Answer: (a) 1
Short Trick: \(\tan kx\sim kx\Rightarrow (3x-2x)/x=1\).
Q67.

\( \displaystyle \lim_{x\to0}\frac{\log(1+4x)}{2x} \)

(a) 0

(b) 1

(c) 2

(d) 4

Answer: (c) 2
Short Trick: \(\log(1+4x)\sim 4x\Rightarrow 4x/(2x)=2\).
Q68.

\( \displaystyle \lim_{x\to0}\frac{e^{3x}-1}{\sin2x} \)

(a) 3/2

(b) 2/3

(c) 1

(d) 0

Answer: (a) 3/2
Short Trick: \(e^{3x}-1\sim 3x\), \(\sin2x\sim 2x\).
Q69.

\( \displaystyle \lim_{x\to0}\frac{\tan5x}{\tan2x} \)

(a) 5/2

(b) 2/5

(c) 1

(d) 0

Answer: (a) 5/2
Short Trick: \(\tan kx\sim kx\Rightarrow 5/2.\)
Q70.

\( \displaystyle \lim_{x\to0}\frac{\sin7x}{x} \)

(a) 7

(b) 1

(c) 0

(d) 1/7

Answer: (a) 7
Short Trick: \(\sin kx/x\to k\).
Q71.

\( \displaystyle \lim_{x\to0}\frac{e^{2x}-e^{x}}{x} \)

(a) \(e^x\)

(b) 1

(c) e

(d) 0

Answer: (b) 1
Short Trick: Expand difference: \((1+2x)-(1+x)=x\).
Q72.

\( \displaystyle \lim_{x\to0}\frac{\sin4x-\sin3x}{x} \)

(a) 1

(b) -1

(c) 1/2

(d) 0

Answer: (a) 1
Short Trick: Formula ⇒ \(2\cos(3.5x)\sin(0.5x)\sim x\).
Q73.

\( \displaystyle \lim_{x\to0}\frac{e^{x}-\cos x}{x^2} \)

(a) 0

(b) 1

(c) 1/2

(d) 3/2

Answer: (d) 3/2
Short Trick: Series → \(x + x^2\) in numerator; divide by \(x^2\to 3/2\).
Q74.

\( \displaystyle \lim_{x\to0}\frac{\tan x – \sin x}{x^3} \)

(a) 0

(b) 1/2

(c) 1/3

(d) 2/3

Answer: (b) 1/2
Short Trick: Same as Q53.
Q75.

\( \displaystyle \lim_{x\to0}\frac{e^{x}-e^{-x}-2x}{x^3} \)

(a) 0

(b) 1/3

(c) 1

(d) 1/6

Answer: (b) 1/3
Short Trick: \(e^{x}-e^{-x}=2x+\tfrac{x^3}{3}+\cdots\). Subtract \(2x\); divide by \(x^3\).\)

Done ✅ Q26–Q75 interactive quiz तैयार है। ऊपर की स्क्रिप्ट सभी “Show Answer” बटन के लिए एक ही है — कोई प्लगइन जरूरी नहीं।


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