If $f(x)=\frac{(81)^{x}-(9)^{x}}{(k)^{x}-1}$ if $x
eq 0$ $=2 \quad$ if $\quad x=0$ is continuous at $x=0$, then the value of $k$ is

Question

If $f(x)=\frac{(81)^{x}-(9)^{x}}{(k)^{x}-1}$ if $x 
eq 0$ $=2 \quad$ if $\quad x=0$ is continuous at $x=0$, then the value of $k$ is

Aicharya · Accepted Answer

$\lim _{x ightarrow 0} \frac{81^{x}-9^{x}}{k^{x}-1}=2$ $	herefore \lim _{x ightarrow 0} \frac{9^{x}\left(9^{x}-1ight)}{k^{x}-1}=2 \Rightarrow \frac{\left(\lim _{x ightarrow 0} 9^{x}ight)\left(\lim _{x ightarrow 0} \frac{9^{x}-1}{x}ight)}{\lim _{x ightarrow 0} \frac{k^{x}-1}{x}}=2$ $\frac{\log 9}{\log k}=2 \Rightarrow \log 9=2 \log k \Rightarrow \log k^{2}=\log 9 \Rightarrow k^{2}=9 \Rightarrow k=3$

If $f(x)=\frac{(81)^{x}-(9)^{x}}{(k)^{x}-1}$ if $x \neq 0$ $=2 \quad$ if $\quad x=0$ is continuous at $x=0$,…

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