Some Standard Limits Involving Logarithmic and Exponential Functions

If $\underset{x\to a}{\lim }f\left(x\right)$ exists, then for finding its value, the value of $f\left(a\right)$ must be known.

For any real number $a(\ne 0), \underset{x\to 0}{\lim }\frac{a^x-1}{x}={\log }_{e}a$.

For any real number $a$ with $a>0$ and $a\ne 1$, $\underset{x\to 0}{\lim }\frac{1}{x}{\log }_a(1+x)=1$

$\underset{x\to 0^{+}}{\lim }{e}^{-\frac{1}{x}}=0$

$\underset{x\to a}{\lim }\log \left(f\left(x\right)\right)=\log \left[\underset{x\to a}{\lim }f\left(x\right)\right]$ provided $\underset{x\to a}{\lim }f\left(x\right)$ is

$\underset{x\to 0}{\lim }\frac{1}{x^2}=$

If $\underset{x\to \infty }{\lim }{\left(\frac{x+c}{x-c}\right)}^x=4$, find $c$.

Evaluate : $\underset{x\to \infty }{\lim }{\left(\frac{2x+3}{2x+1}\right)}^{3x+1}$

Evaluate : $\underset{x\to 0}{\lim }\frac{{10}^x-5^x-2^x+1}{x\tan x}$

Evaluate : $\underset{x\to 0}{\lim }\left(\frac{x·2^x-x}{1-\cos x}\right)$

Evaluate : $\underset{x\to 0}{\lim }\left(\frac{{27}^x-9^x-3^x+1}{\sqrt{2}-\sqrt{1+\cos x}}\right)$

Evaluate : $\underset{x\to 0}{\lim }\left(\frac{{15}^x-5^x-3^x+1}{\sqrt{2}-\sqrt{1+\cos x}}\right)$

Evaluate : $\underset{x\to 0}{\lim }\frac{\left(e^{2x}-1\right){\log }_e(1+x)}{x\sin 2x}$

Evaluate : $\underset{x\to 1}{\lim }\frac{a^{\log x}-x}{\log x}$

Evaluate : $\underset{x\to 2}{\lim }\frac{\sin \left(e^{x-2}-1\right)}{\log (x-1)}$

Evaluate : $\underset{x\to 1}{\lim }(1-x){\log }_{x}2$

Evaluate : $\underset{x\to 0}{\lim }\frac{\sin \left(\log \right(1+x\left)\right)}{\log (1+\sin x)}$

Evaluate : $\underset{x\to 0}{\lim }\frac{\left(e^x-1\right)\log (1+x)}{{\sin }^{2}x}$

Evaluate : $\underset{x\to \infty }{\lim }{\left(\frac{2x+3}{2x+1}\right)}^{x+1}$

Evaluate : $\underset{x\to \infty }{\lim }{\left(1+\frac{2}{x}\right)}^x$