Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (-2+x^2-x)/(-2+x+3*x^2)
-
Limit of (-1+sin(x))/cos(x)
-
Limit of (1/4)^x
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Limit of (-1+e^(-x))/x
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Identical expressions
- (- one +e^(-x))/x
- ( minus 1 plus e to the power of ( minus x)) divide by x
- ( minus one plus e to the power of ( minus x)) divide by x
- (-1+e(-x))/x
- -1+e-x/x
- -1+e^-x/x
- (-1+e^(-x)) divide by x
-
Similar expressions
- (-1+e^(x))/x
- (-1-e^(-x))/x
- (1+e^(-x))/x
Limit of the function (-1+e^(-x))/x
The solution
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[src]
/ -x\
|-1 + E |
lim |--------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{-1 + e^{- x}}{x}\right)$$
Limit((-1 + E^(-x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(1 - e^{x}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{-1 + e^{- x}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\left(1 - e^{x}\right) e^{- x}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - e^{x}\right)}{\frac{d}{d x} x e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{e^{x}}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{x e^{x} + e^{x}}\right)$$
=
$$-1$$
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
$$\lim_{x \to 0^+}\left(1 - e^{x}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{-1 + e^{- x}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\left(1 - e^{x}\right) e^{- x}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - e^{x}\right)}{\frac{d}{d x} x e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{e^{x}}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{x e^{x} + e^{x}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{x e^{x} + e^{x}}\right)$$
=
$$-1$$
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
$$\lim_{x \to 0^-}\left(\frac{-1 + e^{- x}}{x}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{-1 + e^{- x}}{x}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{-1 + e^{- x}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{-1 + e^{- x}}{x}\right) = - \frac{-1 + e}{e}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{-1 + e^{- x}}{x}\right) = - \frac{-1 + e}{e}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{-1 + e^{- x}}{x}\right) = -\infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{-1 + e^{- x}}{x}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{-1 + e^{- x}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{-1 + e^{- x}}{x}\right) = - \frac{-1 + e}{e}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{-1 + e^{- x}}{x}\right) = - \frac{-1 + e}{e}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{-1 + e^{- x}}{x}\right) = -\infty$$
More at x→-oo
One‐sided limits
[src]
/ -x\
|-1 + E |
lim |--------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{-1 + e^{- x}}{x}\right)$$
-1
$$-1$$
= -1.0
/ -x\
|-1 + E |
lim |--------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{-1 + e^{- x}}{x}\right)$$
-1
$$-1$$
= -1.0
= -1.0
The graph