ISI2021-MMA: 1

Question

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $f(x) = \frac{2 – \sqrt{x+4}}{\sin 2x}$ for all $x \neq 0.$ Then the value of $f(0)$ is

• A. $ – \frac{1}{8}$
• B. $\frac{1}{8}$
• C. $0$
• D. $ – \frac{1}{4}$

Answer 1

Ans:1/8

Lim x->0 2-√x+4/sin2x

Lim x->0 1/2√x+4*2cos2x=1/8

Comments

Did not get the answer what happened.Can you explain the answer clearly ?

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he found the limit

Answer 2

Ans: option A (-1/8)

f(0) = lim_x->0 (2-√(x+4))/(sin(2x)) 

Since we have an indeterminate form of type 0/0, we can apply the l'Hopital's rule:

f(0) = lim_x->0 (-1/2)*(1/√(x+4))/(2cos(2x))

now, putting x=0 -->

f(0) = (-1/2)*(1/2)*(1/2)

f(0)= (-1/8)

option(A) - 1/8