Asymptotes to tanhI am new to this site. I have just started reading hyperbolic functions in calculus. I wanted to know if we could derive the shape of y = tanh ⁡ x using derivative tests ( if yes, how?). By shape I mean,It has two horizontal asymptotes at x=1, x=−1.it is inverted u in shape when x belongs to R + It is U shaped when x belongs to R −

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Asymptotes to tanhI am new to this site. I have just started reading hyperbolic functions in calculus. I wanted to know if we could derive the shape of   y  =  tanh  ⁡  x using derivative tests ( if yes, how?). By shape I mean,It has two horizontal asymptotes at x=1, x=−1.it is inverted u in shape when x belongs to             R      +      It is U shaped when x belongs to             R              −

Bryce Obrien · Accepted Answer

The asymptotes   y  =  ±  1 are immediate from   tanh  ⁡  x  =                    e        x            +              e                  −          x                                    e        x            −              e                  −          x                     by noticing that the first terms dominate towards   +  ∞ giving             1      +      0              1      −      0        =  +  1 and the second terms dominate towards   −  ∞ giving             0      +      1              0      −      1        =  −  1. I&#039;m not sure you can use a derivative test to establish this, though.For the others, just note that  (  tanh  ⁡  x      )    ′    =      1                  cosh        2            ⁡      x        (  tanh  ⁡  x      )    ″    =            −      2      sinh      ⁡      x                      cosh        3            ⁡      x      Recall that coshx is always positive, and that sinhx has the sign of x. Thus:-- The first derivative is always positive (so the function is strictly increasing everywhere).-- The second derivative is positive for x&amp;lt;0 (so the function is concave up there), and is negative for x&amp;gt;0 (so the function is concave down there).That&#039;s pretty much everything you mentioned, right?

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