Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (5-4*x+3*x^2)/(1-x+2*x^2)
-
Limit of ((1+2*x)/(-1+x))^(4*x)
-
Limit of (4+x^2-5*x)/(-16+x^2)
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Limit of sinh(x)/x
- Integral of d{x}:
-
sinh(x)/x
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Identical expressions
- sinh(x)/x
- hyperbolic sinus of e of (x) divide by x
- sinhx/x
- sinh(x) divide by x
-
Similar expressions
- sinh(x)/(x*(5+x^2+2*x))
- (-sinh(a)+sinh(x))/(x-a)
- (x-sinh(x))/(x^2*sinh(x))
- sinh(x)/(x*(-1+e^x))
Limit of the function sinh(x)/x
The solution
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[src]
/sinh(x)\ lim |-------| x->0+\ 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
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sinh{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sinh{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+} \cosh{\left(x \right)}$$
=
$$\lim_{x \to 0^+} \cosh{\left(x \right)}$$
=
$$1$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sinh{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sinh{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+} \cosh{\left(x \right)}$$
=
$$\lim_{x \to 0^+} \cosh{\left(x \right)}$$
=
$$1$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sinh{\left(x \right)}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{2 e}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{2 e}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{2 e}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \frac{-1 + e^{2}}{2 e}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sinh{\left(x \right)}}{x}\right) = \infty$$
More at x→-oo
One‐sided limits
[src]
/sinh(x)\ lim |-------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sinh{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
/sinh(x)\ lim |-------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\sinh{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
= 1.0
The graph