Exercise 43

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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) \[\lim_{x \to 0} \frac{x-\ln{\left(1+x\right)}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\]