## Limits and Continuity: IITJEE Maths: Tutorial Problem Set 2

Problem 1: Find $\lim_{x \rightarrow \infty}(1+\frac{2}{x})^{x}$.

Problem 2: If $G(x)=-\sqrt{25-x^{2}}$, then what is the value of $\lim_{x \rightarrow 1} \frac{G(x)-G(1)}{x-1}$?

Problem 3: If $f(x) = (1-x)\tan{\frac{\pi}{2}}$, then find the value of $\lim_{x \rightarrow 1}f(x)$

Problem 4: Find the value of $\lim_{x \rightarrow 2}\frac{\sqrt{(x^{2}+5)} -3}{(x-2)}$

Problem 5: Find the value of $\lim_{x \rightarrow \frac{\pi}{4}}(\sin{2x})^{\tan^{2}{2x}}$

Problem 6: Find the value of $\lim_{x \rightarrow 0}\frac{\sin{x} - x}{x^{3}}$

Problem 7: Find the value of $\lim_{n \rightarrow \infty}$

Problem 8: If $m, n \in N$, what is the relationship between m and n if $\lim_{x \rightarrow 0} \frac{(\sin)^{n}(x)}{\sin{(x^{m})}}=0$

Problem 9: Find $\lim_{x \rightarrow 0}\frac{e^{ax}-e^{bx}}{x}$

Problem 10: Find the value of a given the following:

$f(x) = - 4x$ when x less than or equal to -2.

$f(x)=a.x.x$ when x greater than -2.

given that $\lim_{x \rightarrow -2} {f(x)}$ exists.

Problem 11: Let $f(x) = \frac{\sin{(e^{x-2}-1)}}{\log{(x-1)}}$, then find the value of $\lim_{x \rightarrow 2} f(x)$

Problem 12: Let $f(x) = \frac{1}{\sqrt{(18-x*x)}}$ then find the value of $\lim_{x \rightarrow 3}\frac{f(x)-f(3)}{x-3}$

Problem 13: The function $f(x) = \frac{\log{(1+ax)-\log{(1-ax)}}}{x}$ is not defined when x is zero. In order to make this function continuous at zero, what should be the value of f at zero?

Problem 14: If the function given below is continuous at x=3, find the value of c:

$f(x) = 3x-5$ for $x<3$

$f(x)=x+1$ for $x>3$

$f(x)=c$ for $x=3$

Problem 15: If $f(x) = x+2$, when $x \leq 1$ and $f(x)=4x-1$, when $f(x) = 4x-1$ when $x>1$, then which of the following is true ? (a) f(x) is continuous at $x=1$ (b) $\lim_{x \rightarrow 1}f(x) =4$ (c) f(x) is discontinuous at $x=4$ (d) none of these.

Problem 16:

If $\phi(x) = \frac{1-\cos{(\lambda x)}}{x \sin{x}}$, when x is not zero, and $\phi{(0)} = \frac{1}{2}$. If $\phi{(x)}$ is continuous at $x=0$, then find the value of $\lambda$.

Problem 17:

Let $f(x) = \frac{1-\sin{(x)}}{(\pi-2x)^{2}}$ where $x \neq \frac{\pi}{2}$ and $f(\frac{\pi}{2})=\lambda$, then which value of $\lambda$ will make f(x) continuous at $x = \frac{\pi}{2}$

Regards,

Nalin Pithwa

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