Question: Find the following limits

(a) \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{{{e}^{x}}-x-1}\)

(b) \(\underset{t\to 0}{\mathop{\lim }}\,\frac{t-\ln \left( 1+2t \right)}{{{t}^{2}}}\)

(c) \(\underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{n} \right)}^{n}}\)

(d) \(\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{\frac{1}{x}}}\)

(e) \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\pi }^{x}}-{{3}^{x}}}{x}\)

(f) \(\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,{{x}^{1-\cos x}}\)

(g) \(\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\left( \frac{1}{x}-\frac{1}{\ln \left( x+1 \right)} \right)\)