Question

$\underset{x\to 0}{lim}\frac{1-\cos (1-\cos x)}{x^4}=$

• A. $\frac{1}{8}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $1$

Answer 1

Answer: (A)

$\frac{1}{8}$

$\underset{x\to 0}{lim}\frac{1-\cos (1-\cos x)}{x^4}$

$=\underset{x\to 0}{lim}\frac{1-\cos (1-\cos x)}{{(1-\cos x)}^2}\times \frac{{(1-\cos x)}^2}{x^4}$

$\underset{x\to 0}{lim}\frac{1-\cos (1-\cos x)}{{(1-\cos x)}^2}\times \underset{x\to 0}{lim}{\left(\frac{1-\cos x}{x^2}\right)}^2$

As $x\to 0$

$1-\cos x\to 0$

$=\underset{1-\cos x\to 0}{lim}\frac{1-\cos (1-\cos x)}{{(1-\cos x)}^2}\times {\left(\frac{1}{2}\right)}^2$

$=\frac{1}{2}\times {\left(\frac{1}{2}\right)}^2=\frac{1}{8}$