Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of 1-cos(3*x)
-
Limit of ((6+5*x)/(-1+5*x))^((1+2*x^2)/x)
-
Limit of (-3*5^(1+x)+5*2^x)/(2*5^x+100*2^x)
-
Limit of (4^x-9^x)/(4^(1+x)+9^(1+x))
- Graphing y =:
- (-exp(-x)+exp(x))/x
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Identical expressions
- (-exp(-x)+exp(x))/x
- ( minus exponent of ( minus x) plus exponent of (x)) divide by x
- -exp-x+expx/x
- (-exp(-x)+exp(x)) divide by x
-
Similar expressions
- (-exp(-x)-exp(x))/x
- (-exp(x)+exp(x))/x
- (exp(-x)+exp(x))/x
Limit of the function (-exp(-x)+exp(x))/x
The solution
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[src]
/ -x x\
|- e + e |
lim |----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e^{- x}}{x}\right)$$
Limit((-exp(-x) + exp(x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(e^{2 x} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x e^{x}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e^{- x}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\left(e^{2 x} - 1\right) e^{- x}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{2 x} - 1\right)}{\frac{d}{d x} x e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 e^{2 x}}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2}{x e^{x} + e^{x}}\right)$$
=
$$2$$
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
$$\lim_{x \to 0^+}\left(e^{2 x} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(x e^{x}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e^{- x}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\left(e^{2 x} - 1\right) e^{- x}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{2 x} - 1\right)}{\frac{d}{d x} x e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 e^{2 x}}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2}{x e^{x} + e^{x}}\right)$$
=
$$2$$
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
One‐sided limits
[src]
/ -x x\
|- e + e |
lim |----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e^{- x}}{x}\right)$$
2
$$2$$
= 2.0
/ -x x\
|- e + e |
lim |----------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{e^{x} - e^{- x}}{x}\right)$$
2
$$2$$
= 2.0
= 2.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{e^{x} - e^{- x}}{x}\right) = 2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e^{- x}}{x}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{x} - e^{- x}}{x}\right) = \frac{-1 + e^{2}}{e}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e^{- x}}{x}\right) = \frac{-1 + e^{2}}{e}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e^{- x}}{x}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{x} - e^{- x}}{x}\right) = \frac{-1 + e^{2}}{e}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e^{- x}}{x}\right) = \frac{-1 + e^{2}}{e}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e^{- x}}{x}\right) = \infty$$
More at x→-oo
The graph