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List of top Mathematics Questions on Limits asked in WBJEE

Let \( \Gamma \) be the curve \( y = b e^{-x/a} \) and \( L \) be the straight line:
\[ \frac{x}{a} + \frac{y}{b} = 1, \quad a, b \in \mathbb{R}. \]
Then:
Evaluate: $$ \lim_{n \to \infty} \frac{1}{n^{k+1}} \left[ 2^k + 4^k + 6^k + \dots + (2n)^k \right]. $$ 
Consider the function
\[ f(x) = x(x - 1)(x - 2) \cdots (x - 100). \]
Which one of the following is correct?
Let \[ f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2 \sin x & x^3 & 2x \\ \tan x & x & 1 \end{vmatrix}, \] then \[ \lim_{x \to 0} \frac{f(x)}{x^2} = ? \]
Let \(f : \mathbb{R} \to \mathbb{R}\) be given by \(f(x) = |x^2 - 1|\), then:
Let \(y = f(x)\) be any curve on the X-Y plane and \(P\) be a point on the curve. Let \(C\) be a fixed point not on the curve. The length \(PC\) is either a maximum or a minimum. Then:
\(f(x) = \cos x - 1 + \frac{x^2}{2!}, \, x \in \mathbb{R}\)
Then \(f(x)\) is: