Chapter 1.15, Problem 23P

Find the following limits using Maclaurin series and check your results by computer. Hint: First combine the fractions. Then find the first term of the denominator series and the first term 0f the numerator series.

(a) $\underset{x\to 0}{\lim }\left(\frac{1}{x}-\frac{1}{e^x-1}\right)$

(b) $\underset{x\to 0}{\lim }\left(\frac{1}{x^2}-\frac{\cos x}{{\sin }^{2}x}\right)$

(c) $\underset{x\to 0}{\lim }\left({\csc }^{2}x-\frac{1}{x^2}\right)$

(d) $\underset{x\to 0}{\lim }\left(\frac{1n(1+x)}{x^2}-\frac{1}{x}\right)$