Limits, Continuity and Differentiability
79551
\(\lim _{\mathrm{x} \rightarrow \mathrm{a}} \frac{\log (\mathrm{x}-\mathrm{a})}{\log \left(\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{\mathrm{a}}\right)}\) is equal to :
1 0
2 1
3 a
4 does not exist
Explanation:
(B) : Given,
\(\lim _{x \rightarrow a} \frac{\log (x-a)}{\log \left(e^{x}-e^{a}\right)}\)
Using L- Hospital's rule
\(=\lim _{x \rightarrow a} \frac{\left(\frac{1}{x-a}\right)}{\frac{1}{e^{x}-e^{a}} \cdot e^{x}}=\lim _{x \rightarrow a} \frac{e^{x}-e^{a}}{e^{x}(x-a)}\)
Again using L-Hospital's rule
\(=\lim _{x \rightarrow a} \frac{e^{x}-0}{e^{x}(1-0)+(x-a) e^{x}}=\lim _{x \rightarrow a} \frac{e^{x}}{e^{x}+e^{x}(x-a)}\)
\(=\lim _{x \rightarrow a} \frac{e^{x}}{e^{x}(1+x-a)}=\frac{1}{1+a-a}=1\)
