LIMITS AND DERIVATIVES - Pyqs

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Q1
Evaluate \(\lim_{x\to 0}\frac{\sin 5x}{x}\).
(Exam: IIT-JEE Year: 1998)
(A) \(1\)
(B) \(5\)
(C) \(0\)
(D) Does not exist
Q2
Find \(\lim_{x\to 0}\frac{1-\cos x}{x^2}\).
(Exam: IIT-JEE Year: 2001)
(A) \(0\)
(B) \(1\)
(C) \(\tfrac12\)
(D) \(2\)
Q3
If \(f(x)=|x|\), then \(f'(0)\) is
(Exam: AIPMT Year: 2005)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) Not defined
Q4
Evaluate \(\lim_{x\to0}\frac{e^x-1}{x}\).
(Exam: IIT-JEE Year: 2003)
(A) \(0\)
(B) \(1\)
(C) \(e\)
(D) Does not exist
Q5
The derivative of \(x^2\sin x\) at \(x=0\) equals
(Exam: IIT-JEE Year: 2006)
(A) \(0\)
(B) \(1\)
(C) \(2\)
(D) \(-1\)
Q6
If \(f(x)=\tan x\), then \(\lim_{x\to0}f(x)\sin x\) equals
(Exam: BITSAT Year: 2010)
(A) \(0\)
(B) \(1\)
(C) \(\infty\)
(D) \(-1\)
Q7
Evaluate \(\lim_{x\to0}\frac{\log(1+x)}{x}\).
(Exam: IIT-JEE Year: 1999)
(A) \(0\)
(B) \(1\)
(C) \(e\)
(D) \(-1\)
Q8
The function \(f(x)=x^{1/3}\) is differentiable at \(x=0\)?
(Exam: KVPY Year: 2012)
(A) Yes
(B) No
(C) Only left differentiable
(D) Only right differentiable
Q9
Evaluate \(\lim_{x\to0}\frac{\tan x-x}{x^3}\).
(Exam: IIT-JEE Year: 2004)
(A) \(0\)
(B) \(\tfrac13\)
(C) \(\tfrac12\)
(D) \(1\)
Q10
If \(y=\sin^{-1}x\), then \(\frac{dy}{dx}\) at \(x=0\) equals
(Exam: AIIMS Year: 2008)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) \(\infty\)
Q11
Evaluate \(\lim_{x\to a}\frac{x^2-a^2}{x-a}\).
(Exam: IIT-JEE Year: 1995)
(A) \(0\)
(B) \(a\)
(C) \(2a\)
(D) \(a^2\)
Q12
If \(f(x)=\begin{cases}x,&x\ge0\\-x,&x<0\end{cases}\), then \(f'(0)\) is
(Exam: NEET Year: 2016)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) Not defined
Q13
Evaluate \(\lim_{x\to0}\frac{\sin x}{x+\tan x}\).
(Exam: IIT-JEE Year: 2007)
(A) \(0\)
(B) \(\tfrac12\)
(C) \(1\)
(D) \(2\)
Q14
The derivative of \(\ln x\) at \(x=1\) equals
(Exam: BITSAT Year: 2011)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) \(e\)
Q15
Evaluate \(\lim_{x\to0}\frac{a^x-1}{x}\).
(Exam: IIT-JEE Year: 2002)
(A) \(1\)
(B) \(a\)
(C) \(\ln a\)
(D) \(0\)
Q16
Evaluate \(\lim_{x\to0}\frac{\sin ax}{\sin bx}\).
(Exam: IIT-JEE Year: 1997)
(A) \(0\)
(B) \(\frac ab\)
(C) \(\frac ba\)
(D) \(1\)
Q17
If \(f(x)=x^3\), then \(f'(2)\) equals
(Exam: AIPMT Year: 2009)
(A) \(6\)
(B) \(8\)
(C) \(12\)
(D) \(4\)
Q18
Evaluate \(\lim_{x\to0}\frac{e^{2x}-1}{x}\).
(Exam: BITSAT Year: 2013)
(A) \(1\)
(B) \(2\)
(C) \(e\)
(D) \(0\)
Q19
The function \(f(x)=|x-1|\) is differentiable at
(Exam: NEET Year: 2018)
(A) \(x=0\)
(B) \(x=1\)
(C) \(x=2\)
(D) All real \(x\)
Q20
Evaluate \(\lim_{x\to0}\frac{x-\sin x}{x^3}\).
(Exam: IIT-JEE Year: 2008)
(A) \(\tfrac16\)
(B) \(\tfrac13\)
(C) \(0\)
(D) \(1\)
Q21
If \(y=\ln(\sin x)\), then \(\frac{dy}{dx}\) equals
(Exam: IIT-JEE Year: 2010)
(A) \(\cot x\)
(B) \(\tan x\)
(C) \(\csc x\)
(D) \(\sec x\)
Q22
Evaluate \(\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}\).
(Exam: AIPMT Year: 2007)
(A) \(1\)
(B) \(\tfrac12\)
(C) \(0\)
(D) \(2\)
Q23
The derivative of \(\tan x\) at \(x=0\) equals
(Exam: AIIMS Year: 2006)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) Does not exist
Q24
Evaluate \(\lim_{x\to0}\frac{\sin^2 x}{x^2}\).
(Exam: IIT-JEE Year: 1996)
(A) \(0\)
(B) \(1\)
(C) \(2\)
(D) \(\tfrac12\)
Q25
If \(f(x)=x|x|\), then \(f'(0)\) is
(Exam: KVPY Year: 2014)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) Not defined
Q26
Evaluate \(\lim_{x\to0}\frac{\tan x}{x}\).
(Exam: IIT-JEE Year: 1994)
(A) \(0\)
(B) \(1\)
(C) \(\infty\)
(D) \(-1\)
Q27
The derivative of \(e^{x^2}\) is
(Exam: NEET Year: 2019)
(A) \(2xe^{x^2}\)
(B) \(e^{2x}\)
(C) \(xe^x\)
(D) \(2e^x\)
Q28
Evaluate \(\lim_{x\to0}\frac{\cos x-\cos 2x}{x^2}\).
(Exam: IIT-JEE Year: 2009)
(A) \(0\)
(B) \(\tfrac32\)
(C) \(1\)
(D) \(-\tfrac32\)
Q29
If \(y=\tan^{-1}x\), then \(\frac{dy}{dx}\) at \(x=1\) equals
(Exam: BITSAT Year: 2012)
(A) \(1\)
(B) \(\tfrac12\)
(C) \(2\)
(D) \(\tfrac1{\sqrt2}\)
Q30
Evaluate \(\lim_{x\to0}x\ln x\).
(Exam: IIT-JEE Year: 2000)
(A) \(1\)
(B) \(0\)
(C) \(-\infty\)
(D) \(\infty\)
Q31
The derivative of \(\sin x\) at \(x=\pi\) equals
(Exam: NEET Year: 2020)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) \(-\cos\pi\)
Q32
Evaluate \(\lim_{x\to0}\frac{\sin 3x}{x}\).
(Exam: IIT-JEE Year: 1993)
(A) \(1\)
(B) \(3\)
(C) \(0\)
(D) \(9\)
Q33
If \(f(x)=\frac1x\), then \(f'(x)\) equals
(Exam: AIIMS Year: 2004)
(A) \(-\frac1{x^2}\)
(B) \(\frac1{x^2}\)
(C) \(-x^2\)
(D) \(0\)
Q34
Evaluate \(\lim_{x\to0}\frac{\ln(1+x)-x}{x^2}\).
(Exam: IIT-JEE Year: 2011)
(A) \(0\)
(B) \(-\tfrac12\)
(C) \(\tfrac12\)
(D) \(1\)
Q35
The function \(f(x)=x^2\) is differentiable at
(Exam: State Engg. Exam Year: 2015)
(A) \(x=0\)
(B) \(x=1\)
(C) All real \(x\)
(D) No real \(x\)
Q36
Evaluate \(\lim_{x\to0}\frac{a^x-b^x}{x}\).
(Exam: IIT-JEE Year: 2005)
(A) \(a-b\)
(B) \(\ln a-\ln b\)
(C) \(0\)
(D) \(\ln\frac ab\)
Q37
The derivative of \(\cos x\) is
(Exam: NEET Year: 2017)
(A) \(\sin x\)
(B) \(-\sin x\)
(C) \(-\cos x\)
(D) \(\tan x\)
Q38
Evaluate \(\lim_{x\to0}\frac{\sin x-x\cos x}{x^3}\).
(Exam: IIT-JEE Year: 2012)
(A) \(0\)
(B) \(\tfrac13\)
(C) \(\tfrac12\)
(D) \(1\)
Q39
If \(y=\sqrt{x}\), then \(\frac{dy}{dx}\) equals
(Exam: AIPMT Year: 2006)
(A) \(\tfrac1{2\sqrt{x}}\)
(B) \(2\sqrt{x}\)
(C) \(\sqrt{x}\)
(D) \(x\)
Q40
Evaluate \(\lim_{x\to0}\frac{1-\cos 2x}{x^2}\).
(Exam: IIT-JEE Year: 1998)
(A) \(1\)
(B) \(2\)
(C) \(0\)
(D) \(4\)
Q41
The derivative of \(\sec x\) equals
(Exam: BITSAT Year: 2014)
(A) \(\sec x\tan x\)
(B) \(\tan x\)
(C) \(\sec^2 x\)
(D) \(-\sec x\tan x\)
Q42
Evaluate \(\lim_{x\to0}\frac{\tan 2x}{\tan 3x}\).
(Exam: IIT-JEE Year: 2001)
(A) \(\tfrac23\)
(B) \(\tfrac32\)
(C) \(1\)
(D) \(0\)
Q43
If \(f(x)=x^3-3x\), then \(f'(0)\) equals
(Exam: NEET Year: 2021)
(A) \(0\)
(B) \(-3\)
(C) \(3\)
(D) \(6\)
Q44
Evaluate \(\lim_{x\to0}\frac{e^x-\cos x}{x}\).
(Exam: IIT-JEE Year: 2014)
(A) \(0\)
(B) \(1\)
(C) \(\infty\)
(D) Does not exist
Q45
The derivative of \(\ln(\ln x)\) equals
(Exam: KVPY Year: 2016)
(A) \(\frac1{\ln x}\)
(B) \(\frac1{x\ln x}\)
(C) \(\ln x\)
(D) \(\frac1x\)
Q46
Evaluate \(\lim_{x\to0}\frac{\sin x+\tan x}{x}\).
(Exam: IIT-JEE Year: 2007)
(A) \(1\)
(B) \(2\)
(C) \(0\)
(D) \(\infty\)
Q47
The function \(f(x)=|x|\) is continuous at
(Exam: NEET Year: 2015)
(A) \(x=0\)
(B) \(x=1\)
(C) All real \(x\)
(D) No real \(x\)
Q48
Evaluate \(\lim_{x\to0}\frac{\ln(1+2x)}{x}\).
(Exam: IIT-JEE Year: 1999)
(A) \(1\)
(B) \(2\)
(C) \(0\)
(D) \(-1\)
Q49
The derivative of \(x\sin x\) at \(x=0\) equals
(Exam: AIIMS Year: 2010)
(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) Does not exist
Q50
Evaluate \(\lim_{x\to0}\frac{\sin x}{\sqrt{x}}\).
(Exam: IIT-JEE Year: 2013)
(A) \(0\)
(B) \(1\)
(C) \(\infty\)
(D) Does not exist

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