The function f(x)={\\begin{array}{c}\\\{e^{3 x\end{array}-e^{-5 x}}/{x}, , \text{ if } x \neq 0 ; k , \text{ if } x=0}. is continuous at x=0 for the value of k, as

CBSE Class 12 Math 2022 Term I Solved Paper - Question 12

Question : 12 of 50

Marks: +1, -0

The function

f(x)={\{\table \;{e^{3 x}-e^{-5 x}}/{x}, , \;\text " if "\; x ≠ 0 ; k , \;\text " if "\; x=0}.

is continuous at

x=0

for the value of

k

, as

Solution:

Explanation: Since,

f(x)

is continuous at

x=0

, then

{\LHL}= {\RHL}=f(0) \;\text " or "\; {\LHL}= {\RHL}=k

Now,

{\LHL}=\lim ↙{h → 0} \;{e^{3(0-h)}-e^{-5(0-h)}}/{0-h}

=\lim ↙{h → 0} \;{e^{-3 h}-e^{5 h}}/{-h}

=\lim ↙{h → 0} (\;{e^{-3 h}-1}/{-h}) +\lim ↙{h → 0} (\;{e^{5 h}-1}/{h})

=3 \lim ↙{h → 0} (\;{e^{-3 h}-1}/{-3 h}) +5 \lim ↙{h → 0} (\;{e^{5 h}-1}/{5 h})

=3 × 1+5 × 1=8

\;\text " Thus, "\; \;\; k=8 \;\text ". "\;

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