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

The value of $\lim_{x\rightarrow 0}\frac{\sqrt{1+3x}-\sqrt{1-3x}}{x}$ is
Let $y = \dfrac{3x^3 - 2x^2 + x}{|x|}, \; x \ne 0$. Then $\dfrac{dy}{dx}$ at $x=-2$ is equal to:
The value of $\lim_{x \to 1} \dfrac{x - 1}{3\sqrt{x} - 1}$ is equal to:
If \([x]\) denotes the greatest integer less than or equal to \(x\) for \(x \in \mathbb{R}\), then the value of \(\lim_{x \to 0^{+}} \left[ 2[x] - \frac{x}{|x|} \right]\) is equal to
\(\lim_{x \to 11} \frac{x - 11}{\sqrt{x^2 + 48} - 13} =\)
\( \lim_{x \to 0} \frac{\sqrt{\cos 2x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \) is equal to
\( \lim_{x \to 0} \frac{x^2}{\sqrt{2} - \sqrt{1 + \cos x}} \) is equal to

The value of 

\[ \lim_{x \to \infty} \frac{x^3 \sin\left(\frac{1}{x}\right) - 2x^2}{1 + 3x^2} \]

is ______.

Let \( f(x) = \sin x - \cos x \). Then the value of 

\[ \lim_{x \to \frac{\pi}{2}} \frac{f(x) - f\left(\frac{\pi}{2}\right)}{x - \frac{\pi}{2}} \]

is ______.

The value of \( \lim_{n\to\infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{6n} \right] \) is
If \( f(x) = x^6 + 6^x \), then \( f'(x) \) is equal to
\( \lim_{x \to 0} \frac{\sqrt{1+2x} - 1}{x} \) is
\( \lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9}{4x^3 + 9x - 2} \) is equal to
\( \lim_{x\to0} \frac{\int_0^{x^2} \sin(\sqrt{t}) \, dt}{x^2} \)
\( \lim_{x\to0} \frac{1+x-e^x}{x^2} \)
\( \lim_{x\to\infty} \left(\sqrt{x^2+1} - \sqrt{x^2-1}\right) \)
If \( f \) is differentiable and \( \lim_{h\to0} \frac{f(1+h)-f(1)}{h}=5 \), find \( f'(1) \)
If \( f(x) = x^6 + 6^x \), then \( f'(x) \) is equal to
\( \lim_{x \to 0} \frac{\sqrt{1+2x} - 1}{x} \) is
\( \lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9}{4x^3 + 9x - 2} \) is equal to
\( \lim_{x\to0} \frac{1+x-e^x}{x^2} \)
If \( f(x) = x^6 + 6^x \), then \( f'(x) \) is equal to
\( \lim_{x \to 0} \frac{\sqrt{1+2x} - 1}{x} \) is
\( \lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9}{4x^3 + 9x - 2} \) is equal to
If \( f(x) = \sqrt{\frac{x - \sin x}{x + \cos^2 x}} \), then \( \lim_{x \to \infty} f(x) \) is equal to