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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