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

Limit of the function sinh(x)/x

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

i.e. limit for the numerator is
lim⁡x→0+sinh⁡(x)=0\lim_{x \to 0^+} \sinh{\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+(sinh⁡(x)x)\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right)
=
lim⁡x→0+(ddxsinh⁡(x)ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sinh{\left(x \right)}}{\frac{d}{d x} x}\right)
=
lim⁡x→0+cosh⁡(x)\lim_{x \to 0^+} \cosh{\left(x \right)}
=
lim⁡x→0+cosh⁡(x)\lim_{x \to 0^+} \cosh{\left(x \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
02468-8-6-4-2-101002000
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Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(sinh⁡(x)x)=1\lim_{x \to 0^-}\left(\frac{\sinh{\left(x \right)}}{x}\right) = 1
More at x→0 from the left
lim⁡x→0+(sinh⁡(x)x)=1\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) = 1
lim⁡x→∞(sinh⁡(x)x)=∞\lim_{x \to \infty}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \infty
More at x→oo
lim⁡x→1−(sinh⁡(x)x)=−1+e22e\lim_{x \to 1^-}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{2 e}
More at x→1 from the left
lim⁡x→1+(sinh⁡(x)x)=−1+e22e\lim_{x \to 1^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{2 e}
More at x→1 from the right
lim⁡x→−∞(sinh⁡(x)x)=∞\lim_{x \to -\infty}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \infty
More at x→-oo
One‐sided limits [src] 
     /sinh(x)\
 lim |-------|
x->0+\   x   /
lim⁡x→0+(sinh⁡(x)x)\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) 
1
11 
= 1.0
     /sinh(x)\
 lim |-------|
x->0-\   x   /
lim⁡x→0−(sinh⁡(x)x)\lim_{x \to 0^-}\left(\frac{\sinh{\left(x \right)}}{x}\right) 
1
11 
= 1.0
= 1.0
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function sinh(x)/x
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