Monthly Archives: October 2020

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

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

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

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

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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