Calculus
# Limits of Functions

\[\large \lim_{x \to a} \bigg( 2 - \frac{a}{x} \bigg)^{\tan \frac{\pi x}{2a}} = e^{n}\]

Find approximate value of \(n\).

\[ \large \displaystyle \lim_{n \to \infty} \left(\dfrac{a+\sqrt[n]{b}-1}{a}\right)^{n} \]

Calculate the limit above in terms of \(a\) and \(b\), where \(a\) and \(b\) are constants with \(b\geq0 \).

Let \(f:(1,\infty) \rightarrow (0,\infty)\) be a continuous decreasing function with \[ \large \lim_{x\to\infty} \dfrac{f(4x)}{f(8x)} = \, 1 \]

Then

\[ \large \lim_{x\to\infty} \dfrac{f(6x)}{f(8x)} = \, ?\]

Let \(\{x_n\}\) be a sequence such that \(x_1=1,\ x_nx_{n+1}=2n\) for \(n\ge 1\).

Find \(\displaystyle \lim_{n\to\infty} \dfrac{|x_{n+1}-x_n|}{\sqrt{n}}.\)