Chapter 3.11, Problem 23E

Use the definitions of the hyperbolic functions to find each of the following limits.

(a) $\underset{x\to \infty }{\lim }\tanh x$

(b) $\underset{x\to \infty }{\lim }\tanh x$

(c) $\underset{x\to \infty }{\lim }\sinh x$

(d) $\underset{x\to \infty }{\lim }\sinh x$

(e) $\underset{x\to \infty }{\lim }\text{sech }x$

(f) $\underset{x\to \infty }{\lim }\coth x$

(g) $\underset{\begin{array}{l}x\to 0^{+}\\ x\to \text{U}\end{array}}{\lim }\coth x$

(h) $\underset{\begin{array}{l}x\to 0^{-}\\ x\to \text{U}\end{array}}{\lim }\coth x$

(i) $\underset{x\to -\infty }{\lim }\text{csch }x$

(j) $\underset{x\to \infty }{\lim }\frac{\sinh x}{e^x}$