How to solve : $$\lim_{x \to \infty} (1 + 1/5x)^{2x + 6}$$
I know that
$$\lim_{x \to \infty} (1 + 1/x)^{x} = e$$
$$\lim_{x \to \infty} (1 + 1/x)^{3x} = e^3 $$
I tried to differentiate it
$$(2x + 6) (1 + 1/5x)^{2x + 5}(1/5)$$
How to solve : $$\lim_{x \to \infty} (1 + 1/5x)^{2x + 6}$$
I know that
$$\lim_{x \to \infty} (1 + 1/x)^{x} = e$$
$$\lim_{x \to \infty} (1 + 1/x)^{3x} = e^3 $$
I tried to differentiate it
$$(2x + 6) (1 + 1/5x)^{2x + 5}(1/5)$$
So if you have $\displaystyle \lim_{x\to\infty}(1+\frac{1}{ax})^{bx+c}$, you substitute $\displaystyle y=ax$ to get:
$\displaystyle \lim_{x\to\infty}(1+\frac{1}{ax})^{bx+c}=\lim_{y \to \infty}(1+\frac{1}{y})^{\frac{b}{a}y+c}=((\lim_{y \to \infty}(1+\frac{1}{y})^y)^{\frac{b}{a}})(\lim_{y \to \infty}(1+\frac{1}{y})^c)=e^\frac{b}{a}$
since as $\displaystyle y$ goes to infinity, $\displaystyle (1+\frac{1}{y})^c$ goes to 1.
- Hollywood