Let f:R R be a function given by f(x)=array|c|c| 1- 2xx^2, & x0 array where , R.

Question

Let $f:R	o R$ be a function given by

$f(x)=\begin{array}{|c|c|} \frac{1-\cos 2x}{{x}^{2}}, & x0 \end{array}$

where $\alpha ,\beta \in R$. If $f$ is continuous at $x=0$, then ${\alpha }^{2}+{\beta }^{2}$ is equal to :

Accepted Answer

$f(x)=\begin{gathered} \frac{1-\cos 2x}{{x}^{2}}, x0 $

$f(x)$ is continuous at $x=0$

$\Rightarrow f(0)=\underset{x	o {0}^{-}}{lim}f(x)=\underset{x	o {0}^{+}}{lim}f(x)$

$ \underset{x	o {0}^{-}}{lim}f(x)=\alpha \ \underset{x	o {0}^{-}}{lim}(\frac{1-\cos 2x}{{x}^{2}})=\alpha \ \Rightarrow \underset{x	o {0}^{-}}{lim}\frac{2{\sin }^{2}h}{{x}^{2}}=\alpha \ \Rightarrow \underset{h	o 0}{lim}\frac{2{\sin }^{2}}{{h}^{2}}h=\alpha \ \Rightarrow \alpha =2 \ \ \underset{x	o {0}^{+}}{lim}f(x)=f(0) \ \Rightarrow \, \underset{x	o {0}^{+}}{lim}\frac{\beta \sqrt{1-\cos x}}{x}=2 \end{gathered}$

$\Rightarrow \underset{h	o 0}{lim}\frac{\beta \sqrt{\frac{1-\cos h}{{h}^{2}}}{h}^{2}}{h}=2$

$\begin{gathered} \Rightarrow \, \frac{\beta }{\sqrt{2}}=2 \ \Rightarrow \, \beta =2\sqrt{2} \ \Rightarrow \, {\alpha }^{2}+{\beta }^{2}=4+8 \ \, =12 \end{gathered}$

$	herefore $ Option (1) is correct

Solutions · 1