## JEE Advanced Archive Limits, Continuity & Differentiability Integer Type Questions with Answer Key

1. Let a, b ∈ R be such that $\lim _{x\rightarrow 0}\dfrac{x^{2}\sin \left( bx\right) }{ax-\sin x}=1$. Then the value of 6(a + b) is. {IIT JEE Advanced 2016}.

Answer: the value of 6(a + b) is 7.

2. Let m and n be two positive integers greater than 1. If $\lim _{a\rightarrow 0}\left( \dfrac{e^{\cos \left( a^{n}\right) }-e}{a^{m}}\right) =-\dfrac{e}{2}$, then the value of m/n is. {IIT JEE Advanced 2015}.

Answer: the value of m/n is 2.

3. Let f: R → R be continuous odd function, which vanishes exactly at one point and f(1) = 1/2. Suppose that F(x) = $\int ^{x}_{-1}f\left( t\right) dt$ for all x in [-1, 2] and G(x) = $\int ^{x}_{-1}t\left| f\left( f\left( t\right) \right) \right| dt$ for all x in [-1, 2]. If $\lim _{x\rightarrow 1}\dfrac{F\left( x\right) }{G\left( x\right) }=\dfrac{1}{14}$, then the value of f(1/2) is. {IIT JEE 2015}.
Answer: the value of f(1/2) is 7.

4. The largest value of the non negative integer a for which $\lim _{x\rightarrow 1}\left( \dfrac{-ax+\sin \left( x-1\right) +a}{x+\sin \left( x-1\right) -1}\right) ^{\left( \dfrac{1-x}{1-\sqrt{x}}\right) }=\dfrac{1}{4}$ is. {IIT JEE 2014}.
Answer: the largest value of the non negative integer a is 0.

5. Let f: R → R and Let g: R → R be given by f(x) = |x| + 1 and g(x) = x² + 1. Let h: R → R be defined by h(x) = $\begin{cases}\max \left\{ f\left( x\right) ,g\left( x\right) \right\} ,x\leq 0\\ \min \left\{ f\left( x\right) ,g\left( x\right) \right\} ,x >0\end{cases}$. Then the number of points where the function h(x) is not differentiable is/are. {IIT JEE 2014}.
Answer: the number of points where the function h(x) is not differentiable are 3.

6. Let f(x) be a twice differentiable function such that f(a) = 0, f(b) = 2, f(c) = -1, f(d) = 2, and f(e) = 0, where a < b < c < d < e. Then the minimum number of zeros of g(x) = (f'(x))² + f"(x)f(x) in the interval [a, e] is/are. {IIT JEE 2006}.
Answer: the minimum number of zeros of g(x) are 6.

7. Let f(x) be a continuous function defined for 1 ≤ x ≤ 3. If f(x) takes rational values for all x in R and f(2) = 10, then the value of f(1.5) is. {IIT JEE 1997}.
Answer: the value of f(1.5) is 10.

8. Let F(x) = f(x)g(x)h(x) for all x in R, where f(x), g(x), and h(x) are differentiable functions. At some point x = a, F'(a) = 21F(a), f'(a) = 4f(a), g'(a) = -7g(a), and h'(a) = kh(a). Then the value of k is. {IIT JEE 1997}.
Answer: the value of k is 24.

9. The value of $\lim _{h\rightarrow 0}\dfrac{\ln \left( 1+2h\right) -2\ln \left( 1+h\right) }{h^{2}}$ is. {IIT JEE 1997}.
Answer: the value is -1.

10. If xeˣʸ = y + sin²x, then the value of dy/dx at x = 0 is. {IIT JEE 1996}.
Answer: the value of dy/dx at x = 0 is 1.

11. Let f(x) = x|x|. The set of point(s) where f(x) is twice differentiable is/are. {IIT JEE 1992}.
Answer: f(x) is differentiable everywhere except at x = 0.

12. If f(x) = |x – 2| and g(x) = f(f(x)), then for x > 20 the value of g'(x) is. {IIT JEE 1990}.
Answer: the value of g'(x) is 1.

13. The value of $\lim _{x\rightarrow \infty }\left( \dfrac{x+6}{x+1}\right) ^{x+4}$ is. {IIT JEE 1990}.
Answer: the value is e⁵.

14. If f(9) = 9, f'(9) = 4, then the value of $\lim _{x\rightarrow 9}\dfrac{\sqrt{f\left( x\right) }-3}{\sqrt{x}-3}$ is. {IIT JEE 1988}.
Answer: the required value is 4.

15. Evaluate $\lim _{x\rightarrow \infty }\left( \dfrac{x^{4}\sin \left( \dfrac{1}{x}\right) +x^{2}}{1+\left| x\right| ^{3}}\right)$. {IIT JEE 1988}.
Answer: the value is -1.

16. If f(x) = $\begin{cases}\sin x,x\neq n\pi ,n\in I\\ 2,otherwise\end{cases}$ and g(x) = $\begin{cases}x^{2}+1,x\neq 0,2\\ 4,x=0\\ 5,x=2\end{cases}$, then the value of $\lim _{x\rightarrow 0}g\left( f\left( x\right) \right)$ is. {IIT JEE 1986}.
Answer: the value is 1.

17. The derivative of sec⁻¹(1/(2x² – 1)) with respect to $\sqrt{1-x^{2}}$ at x = 1/2 is. {IIT JEE 1986}.
Answer: the value of the derivative at x = 1/2 is 4.

### Audience

This IIT JEE Advanced Archive is designed to help the students who are preparing for the one of the toughest entrance exams worldwide (IIT JEE Advanced). With this archive students will get to know the pattern of questions which are usually ask in the exam.

### Prerequisites

This IIT JEE Advanced Archive demands to have the detail knowledge of the chapter “Limits, Continuity, and Differentiability”. The other basic requirement to attempt the questions of this archive is, the students must have solved at least 100 questions of the topic.

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