Limits and Continuity: Part 3: IITJEE maths tutorial problems

Problem 1:

Find the following limit:

\lim_{h \rightarrow 0} 2 \times \frac{\sqrt{3}(\sin{(\frac{\pi}{6}+h)})-\cos{(\frac{\pi}{6}+h)}}{\sqrt{3}h(\sqrt{3}\cos{h}-\sin{h})}

Problem 2:

Let the given function be continuous in the interval [-1,1]. Then what must be the value of p?

f(x) = \frac{\sqrt{(1+px)}-\sqrt{(1-px)}}{x}, when -1 \leq x \leq 0

f(x) = \frac{2x+1}{x-2}, when 0 \leq x \leq 1.

Problem 3:

Let the given function be continuous for 0 \leq x < \infty, then find the most suitable values for a and b:

f(x) = \frac{x^{2}}{a}, for 0 \leq x <1

f(x) = a, for 1 \leq x < \sqrt{2}

f(x) = \frac{2b^{2}-4b}{x^{2}}, for \sqrt{2} \leq x < \infty

Problem 4:

Find the value of the following:

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

Problem 5:

The function f(x) = \frac{1}{x} \times (\sqrt{(1+\sin{x})} - \sqrt{(1-\sin{x})}) is not defined at x=0. The value of f(0) so that f(x) becomes continuous at x=0 is (a) 1 (b) 2 (c) 0 (d) none

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{a^{x}-1}{\sqrt{(1+x)}-1}

Problem 7:

Let the given function be f(x) = \frac{\tan{(\frac{\pi}{4}-x)}}{\cot{(2x)}}. Find the value which should be assigned to f at x = \frac{\pi}{4} so that f is continuous everywhere on the reals.

Problem 8:

Let it be given that n \in N and f(x) = \frac{1+2^{x}+3^{x}+\ldots + n^{x}-n}{x}, x is not zero. What value of f(0) will make the function f continuous on the reals.

Problem 9:

Find the value of the following limit:

\lim_{\theta \rightarrow 0^{+}}\frac{\sin{\sqrt{\theta}}}{\sqrt{(\sin{\theta})}}

Problem 10:

If a = \log_{3}{(3x)} and b = \log_{x}{(3)}, then the find the limiting value of a^{b} as x \rightarrow 1:

Problem 11:

Let it be given that n \in N. Then, the find the value of the following limit:

\lim_{x \rightarrow 0}\frac{\sin{x}+\sin{(2x)}+\ldots + \sin{(nx)}}{\sin{x}+\sin{(3x)}+\sin{(5x)}+\ldots + \sin{(2n-1)x}}

Problem 12:

Let it be given that f(x) = x \sin{(\frac{1}{x})} when x is not zero and f(x) = 0, when x is zero. Then, find the value of the following limit:

\lim_{x \rightarrow 0}f(x).

Problem 13:

Find the value of the following limit:

\lim_{x \rightarrow 0}\frac{e^{x^{2}}-\cos{(x)}}{x^{2}}

Problem 14:

Let it be given that f(x) = \frac{x^{2}-(A+2)x+A}{x-2} when x \neq 2 and f(x) = 2, when x=2 is continuous at x=2. Then, find the value of A.

Regards,

Nalin Pithwa


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