Limits and continuity: part 6: IITJEE math: Tutorial Problems for Practice

Problem 1:

Find the value of the following limit:

\lim_{y \rightarrow x} \frac{y^{y}-x^{x}}{y-x}

Problem 2:

Find the value of the following limit:

\lim_{x \rightarrow 2} \frac{e^{\log {(\frac{3x+4}{x})}} - e^{\log{5}}}{x-2}

Problem 3:

Find the value of the following limit:

\lim_{h \rightarrow 0} \frac{(a+h)^{2} \sin{(a+h)}-a^{2}\sin{a}}{h}. Choose one of the following: (i) a\cos{a} -2 \sin{a} (ii) a \cos{a} +2\sin{a} (iii) a(a\cos{a} + 2 \sin{a}) (iv) a^{2} \cos{a}

Problem 4:

Find the value of the following limit:

\lim_{x \rightarrow -2} \frac{x^{5}+2x^{4}+x^{2}+3x+2}{x^{4}+2x^{3}+3x^{2}-5x-22}

Problem 5:

Find the value of the following limit:

\lim_{x \rightarrow \infty} (\frac{x+1}{x+2})(\frac{2x+1}{3x+4})

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10} + \ldots + (x+100)^{10}}{(x^{10}+10^{10})}

Problem 7:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\cos{(ax)}-\cos{(bx)}}{\cos{(cx)} - \cos{(dx)}}

Problem 8:

Find the value of the following limit:

\lim_{x \rightarrow \pi} \frac{1-\cos{(7(x-\pi))}}{5(x-\pi)^{2}}

Problem 9:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{\sqrt[3]{x^{2})-2.\sqrt[3]{(x)}+1}()}{(x-1)^{2}}

Problem 10:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\sec{(4x)} - \sec{(2x)}}{(\sec{(3x)}-\sec{x})}

Problem 11:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{1+\cos{\pi x}}{\tan^{2}{\pi x}}

Problem 12:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{3 \sin{x\pi } -\sin{3x \pi}}{(x-1)^{3}}

Problem 13:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\tan^{4}{x} - \sin^{4}{x}}{x^{6}}

Problem 14:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{x^{3}\sin{x}}{(\sec{x} - \cos{x})^{2}}

Problem 15:

If the value of the following limit is -1, then find the value of a:

\lim_{x \rightarrow a} \frac{\sin{x} -\sin{a}}{\cos{x} -\cos{a}}

Problem 16:

Find the value of the following limit:

\lim_{x \rightarrow 0} (\tan{(\frac{\pi}{4}-x)})^{\frac{1}{x}}

Problem 17:

Find \lim_{x \rightarrow 0} \frac{f^{'}(x)}{x} if f(x) is given as follows:

f(x) = \left | \begin{array}{ccc} \cos{x} & x & 1 \\ 2\sin{x} & x^{2} & 2x \\ \tan{x} & x & 1 \end{array} \right |

Problem 18:

If f(a) = \lim_{x \rightarrow \infty} x(a^{\frac{1}{x}}-1), then f(ab) is equal to (i) f(a).f(b) (ii) f(a)+f(b) (iii) 0 (iv) ab

Problem 19:

Evaluate the following limit:

\lim_{x \rightarrow 1} (\frac{x^{3}+2x^{2}+x+1}{x^{2}=2x+3})^{\frac{1-\cos{(x-1)}}{(x-1)^{2}}}

Problem 20:

The function f is defined by :

f(x) = \frac{e^{x}+e^{-x}-2}{x \sin{x}} in the interval [\frac{\pi}{2}, - \frac{\pi}{2}]-\{0\}

In order for this function to be continuous in [\frac{\pi}{2}, -\frac{\pi}{2}], we have to define (a) f(0)=2 (b) f(\frac{\pi}{2})=1 (c) f(0)=1 (d) f(-\frac{\pi}{2})=1

Problem 21:

The function f(x) = \frac{|x|}{x} when x \neq 0 and f(0)=0, (a) has removable discontinuity at x=0 (b) has irremovable discontinuity at x=0 (c) is continuous at x=0 (d) \lim_{x \rightarrow 0}f(x) exists.

Problem 22:

Let f(x) be defined by

f(x) = \sin{2x} if 0 < x \leq \frac{\pi}{6}

f(x) = ax + b, if \frac{\pi}{6} < x \leq 1

If f(x) and f^{'}(x) are continuous in (0,1), then the value of b is (i) \frac{1}{2} - \frac{\pi}{6} (ii) \frac{1}{2} - \frac{\pi}{3} (iii) \frac{\sqrt{3}}{2} + \frac{\pi}{6} (iv) \frac{\sqrt{3}}{2} - \frac{\pi}{6}

Problem 23:

If f(x) = x^{\frac{2}{3}}-2, where x \geq 0, then \lim_{x \rightarrow 2} f^{-1}(x) is (a) 7 (b) 8 (c) 9 (d) 10

Problem 24:

If f(x) = \frac{x+1}{x+2} and g(x) = \frac{1}{x} then \lim_{x \rightarrow 2} (f+g)(x) is (a) \frac{1}{5} (b) \frac{2}{5} (c) \frac{3}{5} (d) \frac{4}{5}

Regards,

Nalin Pithwa

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