Let , f(x)=array|c|c| (1+ax)^1/x & ,x0 array be continuous at x=0.

Question

Let $\, f(x)=\begin{array}{|c|c|} (1+ax{)}^{1/x} & ,x0 \end{array}$ be continuous at $x=0$. Then ${e}^{a}bc$ is equal to:

Accepted Answer

$f(x)=\begin{gathered} (1+ax{)}^{1/x} ,x0 \end{gathered}$

$\begin{gathered} f(0)=1+6 \ RHL=\underset{x	o {0}^{+}}{lim}\frac{(x+4{)}^{1/2}-2}{(x+c{)}^{1/3}-2}\, [\frac{0}{0}c=8] \end{gathered}$

$\begin{gathered} =\underset{x	o 0}{lim}\frac{2{(1+\frac{x}{4})}^{1/2}-2}{{(1+\frac{x}{8})}^{1/3}-2} \ =\underset{x	o 0}{lim}\frac{1+\frac{x}{8}-1}{1+\frac{x}{8}\cdot \frac{1}{3}-1}=\frac{\frac{1}{8}}{\frac{1}{8}\cdot \frac{1}{3}}=3 \ 	herefore 1+b=3 \ \Rightarrow b=2 \end{gathered}$

$\begin{gathered} LHL=\underset{x	o 0}{lim}(1+ax{)}^{1/x}=\underset{x	o 0}{lim}{e}^{\frac{1+ax-1}{x}}={e}^{a}=3 \ 	herefore \, {e}^{a}\cdot bc=3\cdot 2.8=48 \end{gathered}$

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