If f ( x ) = { 1 - cos 2 x x 2 , x 0 k , x = 0 , then the value of k which makes the function f continuous at x = 0 is:

Question

If $f (x) =$ ${\begin{matrix} \frac{\sqrt{1 - \cos 2 x}}{x \sqrt{2}}, & x \neq 0 \\ k, & x = 0 \end{matrix},$ then the value of $k$ which makes the function $f$ continuous at $x = 0$ is:

Smart Achievers · Accepted Answer

(1)

Ans. 1

Explanation:

Given that the function $f$ is continuous at $x = 0$ ,

Then, $\lim_{x \to 0} f (x) = k$

$\Rightarrow$ $\lim_{x \to 0} \frac{\sqrt{1 - \cos 2 x}}{x \sqrt{2}} = k$

$\Rightarrow$ $\lim_{x \to 0} \frac{\sqrt{2 \sin^{2} x}}{x \sqrt{2}} = k$

$\Rightarrow$ $\lim_{x \to 0} \frac{\sqrt{2} \sin x}{x \sqrt{2}} = k$

$\Rightarrow$ $\lim_{x \to 0} \frac{\sin x}{x} = k$ $[∴ \lim_{x \to 0} \frac{\sin x}{x} = 1]$

$\Rightarrow$ $1 = k$

Solution

Q.
If $f (x) =$ ${\begin{matrix} \frac{\sqrt{1 - \cos 2 x}}{x \sqrt{2}}, & x \neq 0 \\ k, & x = 0 \end{matrix},$ then the value of $k$ which makes the function $f$ continuous at $x = 0$ is:

1 1

2 - 1

3 0

4 No value

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Solution

Q. If f(x)={1-cos2xx2,x≠0k,x=0, then the value of k which makes the function f continuous at x=0 is:

1 1

2 - 1

3 0

4 No value

Ans. (1) Ans. 1 Explanation: Given that the function f is continuous at x=0, Then, limx→0f(x)=k ⇒ limx→01-cos2xx2=k ⇒ limx→02sin2xx2=k ⇒ limx→02sinxx2=k ⇒ limx→0sinxx=k [∴limx→0sinxx=1] ⇒ 1=k

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Q.
If $f (x) =$ ${\begin{matrix} \frac{\sqrt{1 - \cos 2 x}}{x \sqrt{2}}, & x \neq 0 \\ k, & x = 0 \end{matrix},$ then the value of $k$ which makes the function $f$ continuous at $x = 0$ is: