CBSE Class 11 Mathematics Limits And Derivatives Worksheet Set 03

CBSE Class 11 Mathematics Worksheet - Limits and Derivatives. The questions in the worksheets have been specifically designed by best teachers so that the students can practise them to clear their concepts and get better marks in tests and examinations. Students can download these worksheets and practice them. This will help them to get better marks in examinations. Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

Question. Given \( f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1 \)
Show that \( f'(1) = 100 f'(0) \)
Answer: We have \( f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1 \)
Differentiate w.r.t \( x \)
\( f'(x) = \frac{100x^{99}}{100} + \frac{99x^{98}}{99} + \dots + \frac{2x}{2} + 1 + 0 \)
\( = x^{99} + x^{98} + \dots + x + 1 \)
\( f'(1) = (1)^{99} + (1)^{98} + \dots + 1 + 1 \) (put \( x = 1 \))
\( = 1 + 1 + \dots + 1 + 1 = 100 \)
For \( f'(0) \) put \( x = 0 \)
\( f'(0) = 0 + 0 + \dots + 0 + 1 = 1 \)
L.H.S. \( f'(1) = 100 \)
R.H.S. \( 100 f'(0) = 100 \times 1 = 100 \)
\( \therefore \text{L.H.S.} = \text{R.H.S.} \) ans.

Question. \( y = \sqrt{\frac{1-\cos x}{1+\cos x}} \), find \( \frac{dy}{dx} \)
Answer: We have \( y = \sqrt{\frac{1-\cos x}{1+\cos x}} \)
\( \implies y = \sqrt{\frac{2 \sin^2\left(\frac{x}{2}\right)}{2 \cos^2\left(\frac{x}{2}\right)}} \)
\( \implies y = \sqrt{\tan^2\left(\frac{x}{2}\right)} \)
\( \implies y = \tan\left(\frac{x}{2}\right) \)
Differentiate w.r.t \( x \)
\( \frac{dy}{dx} = \sec^2\left(\frac{x}{2}\right) \cdot \frac{d}{dx}\left(\frac{x}{2}\right) \)
\( \frac{dy}{dx} = \sec^2\left(\frac{x}{2}\right) \times \frac{1}{2} = \frac{1}{2} \sec^2\left(\frac{x}{2}\right) \) ans.

Question. \( f(x) = \frac{\sin(x+a)}{\cos x} \), find \( f^{1}(x) \)
Answer: We have \( f(x) = \frac{\sin(x+a)}{\cos x} \)
Differentiate w.r.t \( x \) (quotient rule)
\( f'(x) = \frac{\cos x \cdot \frac{d}{dx}(\sin(x+a)) - \sin(x+a) \cdot \frac{d}{dx}(\cos x)}{\cos^2 x} \)
\( = \frac{\cos x \cdot \cos(x+a) \cdot \frac{d}{dx}(x+a) - \sin(x+a)(-\sin x)}{\cos^2 x} \)
\( = \frac{\cos x \cdot \cos(x+a)(1) + \sin(x+a)\sin x}{\cos^2 x} \)
\( = \frac{\cos(x+a)\cos x + \sin(x+a)\sin x}{\cos^2 x} \)
\( = \frac{\cos(x+a-x)}{\cos^2 x} \)
\( = \frac{\cos a}{\cos^2 x} \)
\( f'(x) = \cos a \sec^2 x \) ans.

Question. \( \lim_{x \to \infty} \left( \frac{\sqrt{3x^2+1} + \sqrt{2x^2-1}}{4x+3} \right) \)
Divide N & D by \( x \)
Answer: \( = \lim_{x \to \infty} \left( \frac{\frac{\sqrt{3x^2+1}}{x} + \frac{\sqrt{2x^2-1}}{x}}{\frac{4x+3}{x}} \right) \)
\( = \lim_{x \to \infty} \left( \frac{\sqrt{3 + \frac{1}{x^2}} + \sqrt{2 - \frac{1}{x^2}}}{4 + \frac{3}{x}} \right) \)
\( = \frac{\sqrt{3}+\sqrt{2}}{4} \) ans.

Question. Evaluate \( \lim_{x \to \infty} \left(\sqrt{x^2+x+1} - \sqrt{x^2+1}\right) \)
Answer: First make function in fraction by rationalize
\( = \lim_{x \to \infty} (\sqrt{x^2+x+1} - \sqrt{x^2+1}) \frac{(\sqrt{x^2+x+1} + \sqrt{x^2+1})}{(\sqrt{x^2+x+1} + \sqrt{x^2+1})} \)
\( = \lim_{x \to \infty} \left( \frac{x^2+x+1 - (x^2+1)}{\sqrt{x^2+x+1} + \sqrt{x^2+1}} \right) \)
\( = \lim_{x \to \infty} \left( \frac{x}{\sqrt{x^2+x+1} + \sqrt{x^2+1}} \right) \)
Divide N & D by \( x \)
\( = \lim_{x \to \infty} \left( \frac{1}{\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}} + \sqrt{1 + \frac{1}{x^2}}} \right) \)
\( = \frac{1}{\sqrt{1}+\sqrt{1}} = \frac{1}{1+1} = \frac{1}{2} \) ans.

Question. Evaluate \( \lim_{n \to \infty} \left( \frac{1+2+3\dots n}{n^2} \right) \)
Answer: We have \( \lim_{n \to \infty} \left( \frac{1+2+3\dots n}{n^2} \right) \)
\( = \lim_{n \to \infty} \left( \frac{\frac{n(n+1)}{2}}{n^2} \right) \)
\( = \frac{1}{2} \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) \)
\( = \frac{1}{2} (1 + 0) = \frac{1}{2} \) ans.

Question. Evaluate \( \lim_{x \to -\infty} \left(\sqrt{x^2-x+1} + x\right) \)
Answer: Let \( x = -y \)
When \( x \to -\infty \) (limits change)
Then \( y \to \infty \)
\( \implies \lim_{y \to \infty} (\sqrt{y^2 + y + 1} - y) \)
Rationalize & proceed yourself
\( \frac{1}{2} \) ans.

Question. Find the derivative of \( \sqrt[3]{\sin x} \) using first principle method.
Answer: Here \( f(x) = \sqrt[3]{\sin x} = (\sin x)^{\frac{1}{3}} \)
\( f'(x) = \lim_{h \to 0} \left[ \frac{\sin(x+h)^{\frac{1}{3}} - (\sin x)^{\frac{1}{3}}}{h} \right] \)
\( = \lim_{h \to 0} \left[ \frac{\sin(x+h)^{\frac{1}{3}} - (\sin x)^{\frac{1}{3}}}{\sin(x+h) - \sin x} \times \frac{\sin(x+h) - \sin x}{h} \right] \)
When \( h \to 0 \) then \( \sin(x+h) \to \sin x \)
\( = \lim_{\sin(x+h) \to \sin x} \left[ \frac{\sin(x+h)^{\frac{1}{3}} - (\sin x)^{\frac{1}{3}}}{\sin(x+h) - \sin x} \right] \times \lim_{h \to 0} \left[ \frac{\sin(x+h) - \sin x}{h} \right] \)
\( = \frac{1}{3} (\sin x)^{\frac{1}{3} - 1} \times \lim_{h \to 0} \left[ \frac{\sin \frac{h}{2}}{\frac{h}{2}} \right] \times \lim_{h \to 0} \left( \cos \left( \frac{2x+h}{2} \right) \right) \)
\( = \frac{1}{3} \sin^{-\frac{2}{3}} x \cdot \lim_{h \to 0} \left( \frac{\sin \frac{h}{2}}{\frac{h}{2}} \right) \times \lim_{h \to 0} \left( \cos \left( \frac{2x+h}{2} \right) \right) \)
\( = \frac{1}{3} \sin^{-\frac{2}{3}} x \times 1 \times \cos x \)
\( \therefore f'(x) = \frac{1}{3} \sin^{-\frac{2}{3}} x \cdot \cos x \) ans.

Question. Find derivative of \( e^{\sqrt{\tan x}} \) using first principle method.
Answer: \( f(x) = e^{\sqrt{\tan x}} \)
\( f'(x) = \lim_{h \to 0} \left( \frac{e^{\sqrt{\tan(x+h)}} - e^{\sqrt{\tan x}}}{h} \right) \)
\( \implies f'(x) = e^{\sqrt{\tan x}} \lim_{h \to 0} \left( \frac{e^{\sqrt{\tan(x+h)} - \sqrt{\tan x}} - 1}{h} \right) \)
\( = e^{\sqrt{\tan x}} \lim_{h \to 0} \left( \frac{e^{\sqrt{\tan(x+h)} - \sqrt{\tan x}} - 1}{\sqrt{\tan(x+h)} - \sqrt{\tan x}} \times \frac{\sqrt{\tan(x+h)} - \sqrt{\tan x}}{h} \right) \)
When \( h \to 0 \); \( \sqrt{\tan(x+h)} \to \sqrt{\tan x} \)
\( f'(x) = e^{\sqrt{\tan x}} \lim_{\sqrt{\tan(x+h)} \to \sqrt{\tan x}} \left[ \frac{e^{\sqrt{\tan(x+h)} - \sqrt{\tan x}} - 1}{\sqrt{\tan(x+h)} - \sqrt{\tan x}} \right] \times \lim_{h \to 0} \left[ \frac{\sqrt{\tan(x+h)} - \sqrt{\tan x}}{h} \right] \)
\( = e^{\sqrt{\tan x}} \times 1 \cdot \lim_{h \to 0} \left[ \frac{\sqrt{\tan(x+h)} - \sqrt{\tan x}}{h} \times \frac{\sqrt{\tan(x+h)} + \sqrt{\tan x}}{\sqrt{\tan(x+h)} + \sqrt{\tan x}} \right] \)
\( = e^{\sqrt{\tan x}} \times 1 \cdot \lim_{h \to 0} \left[ \frac{\tan(x+h) - \tan x}{h} \times \frac{1}{\sqrt{\tan(x+h)} + \sqrt{\tan x}} \right] \)
\( = e^{\sqrt{\tan x}} \times \lim_{h \to 0} \left[ \frac{\tan h \{1 + \tan(x+h) \tan x\}}{h} \times \frac{1}{\sqrt{\tan(x+h)} + \sqrt{\tan x}} \right] \)
\( = e^{\sqrt{\tan x}} \times \lim_{h \to 0} \left( \frac{\tan h}{h} \right) \times \lim_{h \to 0} \left[ \frac{1 + \tan(x+h) \tan x}{\sqrt{\tan(x+h)} + \sqrt{\tan x}} \right] \)
\( = e^{\sqrt{\tan x}} \times 1 \cdot \left( \frac{1 + \tan^2 x}{\sqrt{\tan x} + \sqrt{\tan x}} \right) \)
\( f'(x) = \frac{e^{\sqrt{\tan x}} \cdot \sec^2 x}{2 \sqrt{\tan x}} \) ans.

Question. If \( y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \infty \). Find \( \frac{dy}{dx} \)
Answer: We have \( y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \infty \) ............. (i)
Differentiate both sides w.r.t \( x \)
\( \frac{dy}{dx} = 0 + \frac{1}{1!} + \frac{2x}{2!} + \frac{3x^2}{3!} + \dots \infty \)
\( \frac{dy}{dx} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots \infty \)
\( \frac{dy}{dx} = y \) ............. From eq. (i) ans.