Exponential Limits Problems and Solutions

Evaluate $\displaystyle \large \lim_{x \,\to\, \infty}{\normalsize \dfrac{e^x}{\Big(1+\dfrac{1}{x}\Big)^{x^2}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{e^x-e^{x\cos{x}}}{x+\sin{x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \Big(1+\sin{x}\Big)^{\Large \frac{1}{x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{(e^{-3x+2}-e^2)\sin{\pi x}}{4x^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \sqrt[x^3]{1-x+\sin{x}}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{2^x-1}{\sqrt{1+x}-1}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize {\Bigg(\dfrac{5x^2+1}{3x^2+1}\Bigg)}^\dfrac{1}{x^2}}$

Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{5^x+5^{-x}-2}{x^2}}$