# Exercise 43

Find: $\lim_{x \to 0} \frac{e^{x}-1-x}{x-\ln{\left(1+x\right)}}$
 Question (i) $\lim_{x \to 0} \frac{x-\sin{x}}{x^{3}}$ Question (ii) $\lim_{x \to 0} \frac{xe^{x}-\ln{1+x}}{x^{2}}$ Question (iii) $\lim_{x \to 0} \frac{e^{2x}-e^{-2x}}{2x}$ Question (iv) $\lim_{x \to 0} \frac{\sqrt{1+x}+\sqrt{1-x}-2}{\ln{\left(1-x^{2}\right)}}$ Question (v) $\lim_{x \to 0} \frac{e^{2x}-1}{\ln{\left(1+x\right)}}$ Question (vi) $\lim_{x \to 0} \frac{x^{2} \tan{x}}{\sqrt{1-x^{2}}-1}$ Question (vii) $\lim_{x \to 0} \frac{\sin^{-1}{x}-x}{x^{3} \cos{x}}$ Question (viii) $\lim_{x \to 0} \left(\frac{1}{\sin{x}}-\frac{1}{x}\right)$ Question (ix) $\lim_{x \to 0} \left(\frac{1}{x}-\cot{x}\right)$ Question (x) $\lim_{x \to 0} \frac{\csc{x}-\cot{x}}{x}$ Question (xi) $\lim_{x \to 0} \frac{\tan{x}-x}{x-\sin{x}}$ Question (xii) $\lim_{x \to 0} \frac{x-\ln{\left(1+x\right)}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}$