Questions & Answers

Question

Answers

A. Strictly increasing

B. Strictly decreasing

C. Strictly increasing in the interval [0,∞) and Strictly decreasing in the interval (-∞,0]

D. Strictly increasing in the interval (-∞,0] and Strictly decreasing in the interval [0,∞)

Answer
Verified

The tanh function or hyperbolic tangent function is the ratio between hyperbolic sine and hyperbolic cosine.

We know that,

$

{\text{sinhx = }}\dfrac{{{{\text{e}}^{\text{x}}}{\text{ - }}{{\text{e}}^{{\text{ - x}}}}}}{{\text{2}}} \\

{\text{coshx = }}\dfrac{{{{\text{e}}^{\text{x}}}{\text{ + }}{{\text{e}}^{{\text{ - x}}}}}}{{\text{2}}} \\

$

Then tanh is given by,

${\text{tanhx = }}\dfrac{{{\text{sinhx}}}}{{{\text{coshx}}}}{\text{ = }}\dfrac{{{{\text{e}}^{\text{x}}}{\text{ - }}{{\text{e}}^{{\text{ - x}}}}}}{{{{\text{e}}^{\text{x}}}{\text{ + }}{{\text{e}}^{{\text{ - x}}}}}}$

For \[{\text{x = 0,tanhx = 0}}\].

As x increases, value of \[{\text{tanhx}}\] also increases,

Also, as x decreases, the value of \[{\text{tanhx}}\] decreases.

So, we can say that the graph of the hyperbolic tangent function is strictly increasing.

Therefore,

$

\mathop {\lim }\limits_{x \to \infty } \tanh x = \mathop {\lim }\limits_{x \to \infty } \dfrac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}} \\

= \dfrac{{\mathop {\lim }\limits_{x \to \infty } \left( {1 - {e^{ - 2x}}} \right)}}{{\mathop {\lim }\limits_{x \to \infty } \left( {1 + {e^{ - 2x}}} \right)}} \\

= \dfrac{{1 - \mathop {\lim }\limits_{x \to \infty } {e^{ - 2x}}}}{{1 + \mathop {\lim }\limits_{x \to \infty } {e^{ - 2x}}}} \\

= \dfrac{1}{1} \\

= 1 \\

$

Similarly, tanh tends to -1 when x tends to negative infinity. So the becomes almost straight at higher and lower values of x.