Limits and Continuity: Part 4: IITJEE Math Tutorial Problems for Practice

Problem 1:

If \alpha, \beta are the two roots of the quadratic equation ax^{2}+bx+c=0, then the find the value of the following limit:

\lim_{x \rightarrow \alpha} \frac{1-\cos{(ax^{2}+bx+c)}}{(x-\alpha^{2})}

Problem 2: Given the following functio; find the value of f(0) so that the function is continuous at zero:

f(x) = \frac{\sqrt{1+x}-(1+x)^{\frac{1}{3}}}{x} when x \neq 0.

Problem 3:

Find the value of the following limit: \lim_{x \rightarrow 0} \frac{\sin {(x \deg)}}{x}

Problem 4:

If \lim_{x \rightarrow 0}\frac{1-\cos{(1-\cos{x})}}{x^{4}}=k, which numerical value divides k^{-2}?

Problem 5:

Find the value of the following limit:

\lim_{x \rightarrow 1} {(\sec{(\frac{\pi x}{2})})(\log{x})}

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow \infty} \frac{(2+x)^{40}(4+x)^{5}}{(2-x)^{45}}

Problem 7:

Let it be given that L = \lim_{x \rightarrow 2} (x^{3}-x^{2}+x-1) and M = \lim_{x \rightarrow -2}(x^{4}-x^{3}+x^{2}-x) then find the value of the following limit:

\lim_{x \rightarrow 1}\frac{Lx^{2}-Mx+2}{Mx^{2}-Lx-2}

Problem 8:

Find the value of the following limit:

\lim_{x \rightarrow \arctan{-3}} \frac{\tan^{2}{x}-2\tan{x}-3}{\tan^{2}-4\tan{x}+3}

Problem 9:

Find the value of the following limit:

\lim_{x \rightarrow \infty} (\frac{x+6}{x+1})^{x+4}

Problem 10:

Find the value of b such that the following function is continuous at every point of its domain:

f(x) = 5x-4, when 0 < x \leq 1, and f(x)=4x^{2}+3bx, when 1 < x < 2.

Problem 11:

Find the value of the following limit:

\lim_{x \rightarrow 0}(\cos{x})^{\frac{1}{x}}

Problem 12:

Find the value of the following limit:

\lim_{h \rightarrow 0} \frac{\tan{(x+h)-\tan{x}}}{x}

Problem 13:

Consider the following: \lim_{x \rightarrow 0}\frac{\sqrt{1-\cos{2x}}}{x}. Then, which of the following is true (a) limit exists and is equal to \sqrt{2} (b) exists and it equals -\sqrt{2} (c) limit does not exist because x - 1 \rightarrow 0 (d) limit does not exist because left hand limit is not equal to right hand limit

Problem 14:

Find the value of the following limit:

\lim_{x \rightarrow \frac{\pi}{4}} (\frac{1-\tan{x}}{1-\sqrt{2}\sin{x}})

Problem 15:

Find the value of p given the following:

\lim_{x \rightarrow 0} \frac{\sin{px}}{\tan{3x}}=4

Problem 16:

The number of points of discontinuity of the function f(x)= \frac{1}{\log{|x|}} is (a) zero (b) 1 (c) 2 (d) 3

Problem 17:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\sin(\pi \cos^{2}{x})}{x^{2}}

Problem 18:

Find the value of the following limit:

\lim_{x \rightarrow 2}\frac{3^{\frac{x}{2}}-3}{3^{x}-9}

Regards,

Nalin Pithwa

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