Limits of Functions

Limits of Functions: Level 4 Challenges


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

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 \[ \large \lim_{x\to\infty} \dfrac{f(4x)}{f(8x)} = \, 1 \]


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

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

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

Compute \[ \large \lim_{ n\to{\infty}}{\dfrac{ 2016(1^{2015}+2^{2015}+ \cdots +n^{2015}) - n^{2016}}{2016(1^{2014}+2^{2014}+\cdots+n^{2014})}}.\]


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