Advertisements
Advertisements
Question
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Advertisements
Solution
\[\lim_{x \to 1} \frac{\sqrt{f\left( x \right)} - 1}{\sqrt{x} - 1}\]
\[ = \lim_{x \to 1} \frac{\sqrt{f\left( x \right)} - 1}{\sqrt{x} - 1} \times \frac{\sqrt{f\left( x \right)} + 1}{\sqrt{f\left( x \right)} + 1} \times \frac{\sqrt{x} + 1}{\sqrt{x} + 1}\]
\[ = \lim_{x \to 1} \frac{\left( f\left( x \right) - 1 \right)\left( \sqrt{x} + 1 \right)}{\left( x - 1 \right)\left( \sqrt{f\left( x \right)} + 1 \right)}\]
\[ = \lim_{x \to 1} \frac{f\left( x \right) - 1}{x - 1} \times \lim_{x \to 1} \frac{\left( \sqrt{x} + 1 \right)}{\left( \sqrt{f\left( x \right)} + 1 \right)}\]
\[ = f'\left( 1 \right) \times \frac{1 + 1}{\sqrt{f\left( 1 \right)} + 1}\]
\[ = 2 \times \frac{2}{1 + 1}\]
\[ = 2\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
Find the derivative of 99x at x = 100.
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(x + a)
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(ax + b) (cx + d)2
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
sinn x
Find the derivative of f (x) = 99x at x = 100
\[\frac{1}{\sqrt{x}}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 2}{3x + 5}\]
(x + 2)3
x ex
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
(2x2 + 1) (3x + 2)
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(x + 2) (x + 3)
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
(ax + b) (a + d)2
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of f(x) = tan(ax + b), by first principle.
(ax2 + cot x)(p + q cos x)
