Question

f(x)={1-coskx /xsinx ,x=0 and 1/2,x=0} find k if limx-0f(x) = f(0)

Answer 1

Please check the attachment. Thank you.

Answer 2

The value of k if $x \stackrel{\lim }{\rightarrow} 0\ f(x)=f(0) \text { is } \mp 1$.  

$f(x)=\frac{1-\cos k x}{x \sin x}$

Applying the formula of trigonometry,

$1-\cos k x=2 \sin ^{2} \frac{b x}{2}$

Putting in the equation, we get

$f(x)=\frac{2 \sin ^{2} \frac{k x}{2}}{x \sin x}$

Given that,

By making the use of some trigonometric formulas. We can solve this question. We didn’t substitute the values directly because it would give us an undefined format of 0/0. Therefore, we will first try to transform this 0/0 format into some other format which is valid and from there we can calculate the value of k, very easily.

$x \stackrel{\lim }{\rightarrow} 0\ f(x)=f(0)$

$\therefore x \stackrel{\lim }{\longrightarrow} 0\left[\frac{2 \sin ^{2} \frac{k x}{2}}{x \sin x}\right]=\frac{1}{2}$

$\Rightarrow 2 x \stackrel{\lim }{\rightarrow} 0\left[\frac{2 \sin ^{2} \frac{k x}{2}}{\frac{k^{2} x^{2}}{4} \times \frac{4}{k^{2} x^{2}}} \cdot \frac{1}{x \frac{\sin x}{x} \times x}\right]=\frac{1}{2}$

$\Rightarrow \frac{2}{4} x \stackrel{\lim }{\rightarrow} 0\left[2 \sin ^{2} \frac{k x}{2}\right] \times x \stackrel{\lim }{\rightarrow} 0\left[\frac{1}{\frac{1}{k^{2} x^{2}} \times \frac{\sin x}{x} \times x^{2}}\right]=\frac{1}{2}$

$\Rightarrow \frac{1}{2} \times(1)^{2} \times \frac{1}{k^{2}} \times \frac{1}{x \stackrel{\lim }{\rightarrow} 0 \frac{\sin x}{x}}=\frac{1}{2}$

$\begin{array}{l}{\Rightarrow \frac{1}{k^{2}} \times 1=1} \\ \\ {\Rightarrow k^{2}=1} \\ \\{\Rightarrow k=\pm 1}\end{array}$