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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
If function $f(x)=\left\{\begin{array}{cl}\frac{e^{3 x}-e^{-5 x}}{x} & , \text { if } x \neq 0 \\ k & , \text { if } x=0\end{array}\right.$ is continuous then value of $k$ :
✓

Answer

(D)
$
\begin{aligned}
\lim _{x \rightarrow 0} f(x) & =\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{-5 x}}{x} \\
& =\lim _{x \rightarrow 0} \frac{\left(e^{3 x}-1\right)+\left(1-e^{-5 x}\right)}{x} \\
& =\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}-\lim _{x \rightarrow 0} \frac{e^{-5 x}-1}{x} \\
& =\lim _{3 x \rightarrow 0} \frac{e^{3 x}-1}{3 x} \cdot 3-\lim _{-5 x \rightarrow 0} \frac{e^{-5 x}-1}{-5 x} \cdot(-5) \\
& =(1)(3)-(1)(-5) \\
& =3+5=8
\end{aligned}
$
for continuity at $x=0$,
$
\begin{aligned}
& & \lim _{x \rightarrow 0} f(x) & =f(0) \\
\Rightarrow & & 8 & =k \\
\therefore & & k & =8
\end{aligned}
$

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