Evaluate the following limits: _ y 0 [ 1-y^2-1+y^2 y^2 ]

Question

Evaluate the following limits: $\lim _{y \rightarrow 0}\left[\frac{\sqrt{1-y^2}-\sqrt{1+y^2}}{y^2}\right]$

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Answer

$\lim _{y \rightarrow 0}\left[\frac{\sqrt{1-y^2}-\sqrt{1+y^2}}{y^2}\right]$
$=\lim _{y \rightarrow 0}\left[\frac{\sqrt{1-y^2}-\sqrt{1+y^2}}{y^2} \times \frac{\sqrt{1-y^2}+\sqrt{1+y^2}}{\sqrt{1-y^2}+\sqrt{1+y^2}}\right]$
$...[$ By rationalization$]$
$=\lim _{y \rightarrow 0} \frac{\left(1-y^2\right)-\left(1+y^2\right)}{y^2\left(\sqrt{1-y^2}+\sqrt{1+y^2}\right)}$
$=\lim _{y \rightarrow 0} \frac{1-y^2-1-y^2}{y^2\left(\sqrt{1-y^2}+\sqrt{1+y^2}\right)}$
$=\lim _{y \rightarrow 0} \frac{-2 y^2}{y^2\left(\sqrt{1-y^2}+\sqrt{1+y^2}\right)}$
$=\lim _{y \rightarrow 0} \frac{-2}{\sqrt{1-y^2}+\sqrt{1+y^2}}$
$\ldots [\because y \rightarrow 0, y \neq 0, \therefore y^2 \neq 0]$
$=\frac{-2}{\sqrt{1-0^2}+\sqrt{1+0^2}}$
$=\frac{-2}{1+1}$
$=-1$

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