Kali ini matjitu.com akan membahas limit fungsi khusus. Secara umum diberikan rumus limit fungsi khusus sebagai berikut :
a. $\lim_{x\rightarrow\infty}\big(1+\frac{n}{x}\big)^{x}=e^{n}$
b. $\lim_{x\rightarrow 0}\big(1+nx\big)^{\frac{1}{x}}=e^{n}$

dengan bilangan euler $e=2,7182818...$

Contoh Soal
1. $\lim_{x\rightarrow\infty}\big(1+\frac{4}{x}\big)^{x}=....$
Jawab :
$\lim_{x\rightarrow\infty}\big(1+\frac{4}{x}\big)^{x}=e^4$

2. $\lim_{x\rightarrow\infty}\big(1-\frac{6}{x}\big)^{x}=....$
Jawab :
$\lim_{x\rightarrow\infty}\big(1+\frac{4}{x}\big)^{x}=e^{-6}$

3. $\lim_{x\rightarrow 0}\big(1+7x\big)^{\frac{1}{x}}=....$
Jawab : $\lim_{x\rightarrow 0}\big(1+7x\big)^{\frac{1}{x}}=e^{7}$

4. $\lim_{x\rightarrow 0}\big(1-11x\big)^{\frac{1}{x}}=....$
Jawab : $\lim_{x\rightarrow 0}\big(1-11x\big)^{\frac{1}{x}}=e^{-11}$

5. $\lim_{x\rightarrow\infty}\big(1+\frac{1}{3x}\big)^{2x}=....$
Jawab :
Misalkan $\frac{1}{3x}=y$, karena $x\rightarrow\infty$ maka $y\rightarrow 0$ sehingga
\begin{align*}
\lim_{x\rightarrow\infty}\big(1+\frac{1}{3x}\big)^{2x}&=\lim_{y\rightarrow 0}\big(1+y\big)^{2\cdot\frac{1}{3y}}\\
&=\lim_{y\rightarrow 0}\big(1+y\big)^{\frac{1}{y}\cdot\frac{2}{3}}\\
&=\bigg(\lim_{y\rightarrow 0}\big(1+y\big)^{\frac{1}{y}}\bigg)^{\frac{2}{3}}\\
&=e^{\frac{2}{3}}
\end{align*}