Question

If $f\left(x\right)=\frac{1-\cos ax}{1-\cos bx}$ for $x\ne 0$, is continuous at $x=0$ then $f\left(0\right)=$

• A. $\frac{a^2}{2}$
• B. $\frac{a}{b^2}$
• C. $\frac{a}{b}$
• D. $\frac{a^2}{b^2}$

Answer 1

Answer: (D)

$\frac{a^2}{b^2}$

Given, $f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\frac{1-\cos ax}{1-\cos bx}$

$=\underset{x\to 0}{lim}\frac{2{\sin }^2\frac{ax}{2}}{2{\sin }^2\frac{bx}{2}}$

$={lim}_{x\to 0}\frac{a^2{\sin }^2\frac{ax}{2}/\left(a^2{x}^2/4\right)}{b^2{\sin }^2\frac{bx}{2}/\left(b^2{x}^2/4\right)}$

$=\frac{a^2}{b^2}$