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

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
\( \lim_{x\to0} \frac{1+x-e^x}{x^2} \)
\( \lim_{x\to0} \frac{\int_0^{x^2} \sin(\sqrt{t}) \, dt}{x^2} \)
\( \lim_{x\to\infty} \left(\sqrt{x^2+1} - \sqrt{x^2-1}\right) \)
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
If \( f:\mathbb{R} \to (0,\infty) \) is an increasing function and if \( \lim_{x \to 2018} \frac{f(3x)}{f(x)} = 1 \), then \( \lim_{x \to 2018} \frac{f(2x)}{f(x)} \) is equal to
If \( f \) is differentiable at \( x = 1 \), then \( \lim_{x \to 1} \frac{x^2 f(1) - f(x)}{x - 1} \) is
\( \lim_{x\to0} \frac{1+x-e^x}{x^2} \)
\( \lim_{x\to0} \frac{\int_0^{x^2} \sin(\sqrt{t}) \, dt}{x^2} \)
\( \lim_{x\to\infty} \left(\sqrt{x^2+1} - \sqrt{x^2-1}\right) \)
If \( f \) is differentiable and \( \lim_{h\to0} \frac{f(1+h)-f(1)}{h}=5 \), find \( f'(1) \)
\( \lim_{x \to 0} \frac{\sqrt{1+2x} - 1}{x} \) is
\( \lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9}{4x^3 + 9x - 2} \) is equal to