Question: Find the following limits

1. \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{{{e}^{x}}-x-1}\)
2. \(\underset{t\to 0}{\mathop{\lim }}\,\frac{t-\ln \left( 1+2t \right)}{{{t}^{2}}}\)
3. \(\underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{n} \right)}^{n}}\)
4. \(\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{\frac{1}{x}}}\)
5. \(\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\pi }^{x}}-{{3}^{x}}}{x}\)
6. \(\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,{{x}^{1-\cos x}}\)
7. \(\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\left( \frac{1}{x}-\frac{1}{\ln \left( x+1 \right)} \right)\)

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