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tanh(x)/x
Limit of the function/ tanh(x)/x

Limit of the function tanh(x)/x

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     /tanh(x)\
 lim |-------|
x->0+\   x   /
lim⁡x→0+(tanh⁡(x)x)\lim_{x \to 0^+}\left(\frac{\tanh{\left(x \right)}}{x}\right) 
Limit(tanh(x)/x, x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+tanh⁡(x)=0\lim_{x \to 0^+} \tanh{\left(x \right)} = 0
and limit for the denominator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(tanh⁡(x)x)\lim_{x \to 0^+}\left(\frac{\tanh{\left(x \right)}}{x}\right)
=
lim⁡x→0+(ddxtanh⁡(x)ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tanh{\left(x \right)}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(1−tanh⁡2(x))\lim_{x \to 0^+}\left(1 - \tanh^{2}{\left(x \right)}\right)
=
lim⁡x→0+(1−tanh⁡2(x))\lim_{x \to 0^+}\left(1 - \tanh^{2}{\left(x \right)}\right)
=
11
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
02468-8-6-4-2-10100.02.0
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Rapid solution [src] 
1
11
Expand and simplify
One‐sided limits [src] 
     /tanh(x)\
 lim |-------|
x->0+\   x   /
lim⁡x→0+(tanh⁡(x)x)\lim_{x \to 0^+}\left(\frac{\tanh{\left(x \right)}}{x}\right) 
1
11 
= 1.0
     /tanh(x)\
 lim |-------|
x->0-\   x   /
lim⁡x→0−(tanh⁡(x)x)\lim_{x \to 0^-}\left(\frac{\tanh{\left(x \right)}}{x}\right) 
1
11 
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(tanh⁡(x)x)=1\lim_{x \to 0^-}\left(\frac{\tanh{\left(x \right)}}{x}\right) = 1
More at x→0 from the left
lim⁡x→0+(tanh⁡(x)x)=1\lim_{x \to 0^+}\left(\frac{\tanh{\left(x \right)}}{x}\right) = 1
lim⁡x→∞(tanh⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\tanh{\left(x \right)}}{x}\right) = 0
More at x→oo
lim⁡x→1−(tanh⁡(x)x)=−1+e21+e2\lim_{x \to 1^-}\left(\frac{\tanh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{1 + e^{2}}
More at x→1 from the left
lim⁡x→1+(tanh⁡(x)x)=−1+e21+e2\lim_{x \to 1^+}\left(\frac{\tanh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{1 + e^{2}}
More at x→1 from the right
lim⁡x→−∞(tanh⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\tanh{\left(x \right)}}{x}\right) = 0
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function tanh(x)/x
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