List of top Mathematics Questions on Limits and Exponential Functions

Let $f(x) = \lim_{y \to 0} \frac{(1 - \cos(xy)) \tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x) = \sin x$, $x \in \mathbb{R}$ is :

Let \[ \lim_{x\to2}\frac{\tan(x-2)\,[x^2+(p-2)x-2p]}{(x-2)^2}=5 \] for some \(p,r\in\mathbb{R}\). If the set of all possible values of \(q\), such that the roots of the equation \(rx^2-px+q=0\) lie in \( (0,2) \), be the interval \( (\alpha,\beta) \), then \(4(\alpha+\beta)\) equals :

The product of all possible values of $\alpha$, for which $\lim_{x \to 0} \frac{1-\cos(\alpha x)\cos((\alpha+1)x)\cos((\alpha+2)x)}{\sin^2((\alpha+1)x)} = 2$, is:

If \[ \lim_{x\to 2}\frac{\sin(x^3-5x^2+ax+b)}{(\sqrt{x-1}-1)\log_e(x-1)}=m, \] then \(a+b+m\) is equal to:

The value of $\lim_{x \to \infty} \left(\frac{x+6}{x+1}\right)^{x+4}$ is:

Evaluate the following limit: \[ \lim_{x \to \infty} \frac{(2x^2 - 3x + 5) \left( 3x - 1 \right)^{x/2}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^x}}. \] The value of the limit is:

If \( \lim_{t \to \infty} \left( \int_0^{1} \left( 3x + 5 \right)^t dx \right) = \frac{\alpha}{5e} \left( \frac{8}{5} \right)^{\frac{3}{2}}, \) then \( \alpha \) is equal to ____ :

The integral value of \(n\) for which \[ \lim_{x\to 0}\frac{(\cos x-1)(\cos x-e^x)}{x^n} \] is a finite non-zero real number is

Evaluate \(\lim_{x \to 0} \frac{x^2 \log(\cos x)}{\log(1+x^2)}\)

Evaluate \[ \lim_{x\to \infty}x^3\left\{\sqrt{x^2+\sqrt{x^4+1}}-\sqrt2x\right\} \]

If \[ \lim_{x\to 0^+}x^2 \left( \frac{e^{1/x}-e^{-1/x}} {e^{1/x}+e^{-1/x}} \right)=k \] and \[ \lim_{x\to 0^-}x^2 \left( \frac{e^{1/x}-e^{-1/x}} {e^{1/x}+e^{-1/x}} \right)=l, \] then

Evaluate \[ \lim_{x\to 0} \frac{\sqrt{11+|x|-6\sqrt{2+|x|}}} {6-2\sqrt{2+|x|}} \]

The value of \[ \lim_{x\to 0}\left(\frac{e^x-1}{x}\right)^{\frac{1+x-e^x}{x}} \] is

If \(a\gt 0,\ b\gt 0\), then \[ \lim_{n\to\infty} \left( \frac{a+b^{1/n}-1}{a} \right)^n = \] is equal to: