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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits2 Marks
Question
Evaluate the following limits:
$\lim _{y \rightarrow 0}\left[\frac{\sqrt{a+y}-\sqrt{a}}{y \sqrt{a+y}}\right]$
✓

Answer

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

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