If fx=(e^x2-cos⁡x)/(x^2), for x≠0 is continuous at x=0, then the value of f0 is

Correct answer: (D) 3/2. Since f( x ) is continuous at x=0 Hence f( 0 )= lim x→0 (e^(x 2) −cosx)/(x^2) = lim x→0 (e^(x 2) −1)/(x^2) + lim x→0 (1−cosx)/(x^2) = lim x → 0 (e^(x 2) - 1)/(x^2) + lim x→0 (2 sin^2 x)/2 x 2^2 × 1/4 =1+(1^2)/2 =3/2

Q99 / 150 · MHT CET 2018 · Mathematics

If $f (x) = \frac{e^{x^{2}} - \cos x}{x^{2}},$ for $x \neq 0$ is continuous at $x = 0$ , then the value of $f (0)$ is

A
$\frac{2}{3}$
B
$\frac{5}{2}$
C
$1$
Correct answer
D
$\frac{3}{2}$

Solution

Since

f (x)

is continuous at

x = 0

Hence

f (0) = \lim_{x \to 0} \frac{e^{x^{2}} - \cos x}{x^{2}}

= \lim_{x \to 0} \frac{e^{x^{2}} - 1}{x^{2}} + \lim_{x \to 0} \frac{1 - \cos x}{x^{2}}

= lim_{x \to 0} (\frac{e^{x^{2}} - 1}{x^{2}}) + \lim_{x \to 0} \frac{2 {sin}^{2} \frac{x}{2}}{{(\frac{x}{2})}^{2}} \times \frac{1}{4}

= 1 + \frac{1^{2}}{2} = \frac{3}{2}

Hence, the correct answer is option: 'D'.

If fx=ex2-cos⁡xx2, for x≠0 is continuous at x=0, then the value of f0 is

Solution

Relevant Questions

If $f (x) = \frac{e^{x^{2}} - \cos x}{x^{2}},$ for $x \neq 0$ is continuous at $x = 0$ , then the value of $f (0)$ is