\(\displaystyle \lim_{x\rightarrow 2} {\left(\frac{2x-3}{x-1}\right)^{\frac{x}{x-2}}} \)
\(\displaystyle \lim_{x\rightarrow 2} {\left(\frac{2x-3}{x-1}\right)^{\frac{x}{x-2}}} = 1^{\infty} \)
Quan \(x \to 2\) el límit presenta una indeterminació \(\displaystyle 1^{\infty}\). Hem de fer servir la definició del nombre \(e\) per calcular el límit.
\(\displaystyle
\begin{align}
\lim_{x\rightarrow 2} {\left(\frac{2x-3}{x-1}\right)^{\frac{x}{x-2}}}
&= \lim_{x\rightarrow 2} {\Bigg(1+\frac{2x-3}{x-1}-1\Bigg)^{\frac{x}{x-2}}} \\[12pt]
&= \lim_{x\rightarrow 2} {\Bigg(1+\frac{2x-3}{x-1}-\frac{x-1}{x-1}\Bigg)^{\frac{x}{x-2}}} \\[12pt]
&= \lim_{x\rightarrow 2} {\Bigg(1+\frac{x-2}{x-1}\Bigg)^{\frac{x}{x-2}}} \\[12pt]
&= \lim_{x\rightarrow 2} {\Bigg(1+\frac{1}{\frac{x-1}{x-2}}\Bigg)^{\frac{x}{x-2}}} \\[12pt]
&= \lim_{x\rightarrow 2} {\Bigg(1+\frac{1}{\frac{x-1}{x-2}}\Bigg)^{\frac{x-1}{x-2} \cdot \frac{x}{x-1}}} \\[12pt]
&= e^{\scriptsize{\displaystyle \left(\lim_{x\rightarrow 2} {\frac{x}{x-1}}\right)}} \\[12pt]
&= e^2
\end{align}
\)