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 (5x3 + 3x – 1) (x – 1).
Find the derivative of x5 (3 – 6x–9).
Find the derivative of x–4 (3 – 4x–5).
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):
`(px^2 +qx + r)/(ax +b)`
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)n
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)n (cx + d)m
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):
cosec x cot x
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
(x2 + 1) (x − 5)
x ex
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan (2x + 1)
\[\sqrt{\tan x}\]
x4 − 2 sin x + 3 cos x
3x + x3 + 33
(2x2 + 1) (3x + 2)
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[\frac{2^x \cot x}{\sqrt{x}}\]
x4 (5 sin x − 3 cos 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.
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
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 x2 cosx.
`(a + b sin x)/(c + d cos x)`
