List of top Mathematics Questions on limits and derivatives asked in BITSAT

Evaluate the following limit: $ \lim_{n \to \infty} \prod_{r=3}^n \frac{r^3 - 8}{r^3 + 8} $.

If $ f(x) $ is defined as follows:
$$ f(x) = \begin{cases} 4, & \text{if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & \text{if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & \text{if } \sqrt{5} \leq x < \infty. \end{cases} $$ If $ k $ is the number of points where $ f(x) $ is not differentiable, then $ k - 2 = $

Let $ f(x) = x^2 \log(\cos x) \log(1 + x) $ for $ x \neq 0 $, and $ f(0) = 0 $. Determine the behavior of $ f(x) $ at $ x = 0 $.

Let $ f $ be the function defined by:
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x^2 - 2|x-1| - 1}, & \text{if } x \neq 1, \\ \frac{1}{2}, & \text{if } x = 1. \end{cases} \] The function is continuous at:

Let the function $ g: (-\infty, 0) \rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $ be given by $ g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} $. Determine the properties of $ g $.

The derivative of $ e^{x^3} $ with respect to $ \log x $ is:

The value of limₓtₒᵢₙfty(fraca₁¹/x+a₂¹/x+⋯+aₙ¹/xn)ⁿx, aᵢ>0 is

If limₓtₒᵢₙfty xsin((1)/(x))=A and limₓ\to0 xsin((1)/(x))=B, then which one of the following is correct?

Evaluate limₓtₒᵢₙftyfracint₀²x x eˣ²dxe⁴x²

The value of
\[ \lim_{n \to \infty} \frac{1 + 2 + 3 + \cdots + n}{n^2 + 100} \]
is equal to:

If $x > 0,$ the $ 1 + \frac{\log_{e^{2}}x}{1!} + \frac{\left(\log_{e^{2}}x\right)^{2}}{2!} + ....=$

The value of
\[ \lim_{x \to \infty} \left(a_1^{1/x} + a_2^{1/x} + \cdots + a_n^{1/x}\right)^{n x}, \quad a_i > 0 \]

The curve $y = xe^x$ has minimum value equal to

At an extreme point of a function $f (x)$, the tangent to the curve is