   Chapter 3.11, Problem 23E

Chapter
Section
Textbook Problem

Use the definitions of the hyperbolic functions to find each of the following limits.(a) lim x → ∞ tanh x (b) lim x → ∞ tanh x (c) lim x → ∞ sinh x (d) lim x → ∞ sinh x (e) lim x → ∞ sech  x (f) lim x → ∞ coth x (g) lim x → 0 +     x → U coth x (h) lim x → 0 −   x → U coth x (i) lim x → − ∞ csch  x (j) lim x → ∞ sinh x e x

(a)

To determine

To find: limxtanhx by using definition of the hyperbolic function.

Explanation

Calculation:

The definition of the hyperbolic function is, tanhx=sinhxcoshx .

Since the definition of the hyperbolic function is, sinhx=exex2 and coshx=ex+ex2.

tanhx=exex2ex+ex2=exexex+ex=e2x1e2x+1=11e2x1+1e2x

(b)

To determine

To find: limxtanhx by using definition of the hyperbolic function.

(c)

To determine

To find: limxsinhx by using definition of the hyperbolic function.

(d)

To determine

To find: limxsinhx by using definition of the hyperbolic function.

(e)

To determine

To find: limxsechx by using definition of the hyperbolic function.

(f)

To determine

To find: limxcothx by using definition of the hyperbolic function.

(g)

To determine

To find: limx0x0cothx use definition of the hyperbolic function.

(h)

To determine

To find: limx0x0cothx use definition of the hyperbolic function.

(i)

To determine

To find: limxcschx by using definition of the hyperbolic function.

(j)

To determine

To find: limxsinhxex use definition of the hyperbolic function.

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