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 \(n\).
by
Mohit Gupta
\[ \large \displaystyle \lim_{n \to \infty} \left(\dfrac{a+\sqrt[n]{b}-1}{a}\right)^{n} \]
Calculate the limit above in terms of \(a\) and \(b\), where \(a\) and \(b\) are constants with \(b\geq0 \).
by
A Former Brilliant Member
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
\[ \large \lim_{x\to\infty} \dfrac{f(6x)}{f(8x)} = \, ?\]
by
Arjun Ramesh
Let \(\{x_n\}\) be a sequence such that \(x_1=1,\ x_nx_{n+1}=2n\) for \(n\ge 1\).
Find \(\displaystyle \lim_{n\to\infty} \dfrac{|x_{n+1}-x_n|}{\sqrt{n}}.\)
by
Khang Nguyen Thanh
Compute \[ \large \lim_{ n\to{\infty}}{\dfrac{ 2016(1^{2015}+2^{2015}+ \cdots +n^{2015}) - n^{2016}}{2016(1^{2014}+2^{2014}+\cdots+n^{2014})}}.\]
by
Ankush Tiwari