$\lim _{x \rightarrow 2}\left(\frac{5^x+5^{3-x}-30}{5^{3-x}-5^{\frac{x}{2}}}\right)=$

Question

Aicharya · Accepted Answer

Let $\begin{aligned} L & =\lim _{x ightarrow 2}\left(\frac{5^x+5^{3-x}-30}{5^{3-x}-5^{\frac{x}{2}}}ight) \ & =\lim _{x ightarrow 2}\left(\frac{5^x+\frac{5^3}{5^x}-30}{\frac{5^3}{5^x}-\left(5^xight)^{\frac{1}{2}}}ight)\end{aligned}$ $\begin{aligned} & 	ext { Let } t=5^x \ 	herefore \quad x & ightarrow 2 \Rightarrow t ightarrow 25 \ 	herefore \quad L & =\lim _{t ightarrow 25}\left(\frac{t+\frac{125}{t}-30}{\frac{125}{t}-\sqrt{t}}ight) \ & =\lim _{t ightarrow 25}\left(\frac{t^2-30 t+125}{25 \sqrt{25}-t \sqrt{t}}ight) \ & =\lim _{t ightarrow 25}\left(\frac{(t-25)(t-5)}{25^{\frac{3}{2}}-t^{\frac{3}{2}}}ight)\end{aligned}$ $\begin{aligned} & =\lim _{\mathrm{t} ightarrow 25} \frac{\mathrm{t}-5}{-\left(\frac{25^{\frac{3}{2}}-\mathrm{t}^{\frac{3}{2}}}{25-\mathrm{t}}ight)} \ & =\frac{25-5}{-\frac{3}{2}(25)^{\frac{1}{2}}} \ & =\frac{-8}{3}\end{aligned}$

$\lim _{x \rightarrow 2}\left(\frac{5^x+5^{3-x}-30}{5^{3-x}-5^{\frac{x}{2}}}\right)=$

Solution