2020 October « Mathematics Hothouse

Monthly Archives: October 2020

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

October 10, 2020 – 4:05 am

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Limits and Continuity: part 5: IITJEE Math: Tutorial problems for practice

October 10, 2020 – 3:16 am

Problem 1:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{\sin{a} - \tan{a}}{\sin^{a}}$

Problem 2:

Find the values of the constant a and b such that the following limit is zero:

$\lim_{x \rightarrow \infty} [\frac{x^{2}+1}{x+1} -ax-b]$

Problem 3:

Find the value of the following limit:

$\lim_{\alpha \rightarrow \beta} \frac{\sin^{2}{\alpha}-\sin^{2}{\beta}}{\alpha^{2}-\beta^{2}}$

Problem 4:

If a, b, c, d are positive, then find the value of the following limit:

$\lim_{x \rightarrow \infty}(1+\frac{1}{a+bx})^{c+dx}$

Problem 5:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{(1-\cos{(2x)})\sin{(5x)}}{x^{2}\sin{(3x)}}$

Problem 6:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} \frac{\sqrt{x^{2}-1}}{2x+1}$

Problem 7:

Find the value of the following limit:

$\frac{\log{(1+x+x^{2})}+\log{(1-x+x^{2})}}{\sec{x}-\cos{x}}$

Problem 8:

Find the value of the following limit:

$\lim_{x \rightarrow \infty} (\frac{2+x}{1+x})^{2x+1}$

Problem 9:

Find the value of f(0) such that the following function is continuous at zero:

$f(x) = (x+1)^{\cot{x}}$

Problem 10:

Let $f^{''}(x)$ be continuous at zero and $f^{''}(0)=4$ . Then, find the numerical value of the following limit:

$\lim_{x \rightarrow 0}\frac{2f(x)-3f(2x)+f(4x)}{x^{2}}$

Problem 11:

Find the value of the following limit:

$\lim_{n \rightarrow \infty} (\frac{n^{3}}{3n^{2}-4} - \frac{n^{2}}{3n+2})$

Problem 12:

Find the values of x where the following function is discontinuous:

$f(x) = \frac{\sin{x} \log{(x-2)}}{(x^{2}-4x+3)}$

Problem 13:

The value of p for which the following function may be continuous at zero is what:

$f(x) = \frac{(4x-1)^{3}}{(\sin{\frac{x}{p}})(\log{(1+\frac{x^{2}}{3})})}$ , when $x \neq 0$ , and

$f(x) = 12(\log{4})^{3}$ , when $x = 0$ .

Problem 14:

Find the value of the following limit:

$\lim_{x \rightarrow 0} \frac{1-\cos{(mx)}}{1-\cos{(nx)}}$

Problem 15:

If $f(x) = \frac{4-7x}{3x+4}$ and $\lim_{x \rightarrow 2}f(x) = k$ , and $\lim_{x \rightarrow 0}f(x) = m$ , then the equation whose roots are $\frac{1}{k}, \frac{1}{m}$ is (a) $x^{2}+x=0$ (b) $x^{2}-1=0$ (c) $x^{2}+1=0$ (d) $x^{2}+2x=0$

Problem 16:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots + x^{n}-n}{x-1}$

Problem 17:

Find the value of the following limit:

$\lim_{x \rightarrow 1} \frac{\sqrt[n]{x^{m}}-1}{\sqrt[m]{x^{n}}-1}$

Problem 18:

Find the value of the following limit:

$\lim_{x \rightarrow a} \frac{\tan{x} - \tan{a}}{\sin{a} - \sin{x}}$

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

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

October 10, 2020 – 2:43 am

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

By Nalin Pithwa | Posted in IITJEE Mains | Comments (0)

Limits and Continuity: Part 3: IITJEE maths tutorial problems

October 9, 2020 – 6:54 pm

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.