Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (e^x-e)/(-1+x)
-
Limit of (1-3*x)^(1/x)
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Limit of pi/(x*cot(pi*x/2))
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Limit of sin(7*x)/x^2
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Identical expressions
- (e^x-e)/(- one +x)
- (e to the power of x minus e) divide by ( minus 1 plus x)
- (e to the power of x minus e) divide by ( minus one plus x)
- (ex-e)/(-1+x)
- ex-e/-1+x
- e^x-e/-1+x
- (e^x-e) divide by (-1+x)
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Similar expressions
- (e^x-e)/(-1-x)
- (e^x-e)/(1+x)
- (e^x+e)/(-1+x)
Limit of the function (e^x-e)/(-1+x)
The solution
You have entered
[src]
/ x \
|E - E|
lim |------|
x->1+\-1 + x/
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)$$
Limit((E^x - E)/(-1 + x), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(e^{x} - e\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(e^{x} - e\right)}{\frac{d}{d x} \left(x - 1\right)}\right)$$
=
$$\lim_{x \to 1^+} e^{x}$$
=
$$\lim_{x \to 1^+} e^{x}$$
=
$$e$$
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
$$\lim_{x \to 1^+}\left(e^{x} - e\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(e^{x} - e\right)}{\frac{d}{d x} \left(x - 1\right)}\right)$$
=
$$\lim_{x \to 1^+} e^{x}$$
=
$$\lim_{x \to 1^+} e^{x}$$
=
$$e$$
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 \
|E - E|
lim |------|
x->1+\-1 + x/
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)$$
E
$$e$$
= 2.71828182845905
/ x \
|E - E|
lim |------|
x->1-\-1 + x/
$$\lim_{x \to 1^-}\left(\frac{e^{x} - e}{x - 1}\right)$$
E
$$e$$
= 2.71828182845905
= 2.71828182845905
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{e^{x} - e}{x - 1}\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right) = e$$
$$\lim_{x \to \infty}\left(\frac{e^{x} - e}{x - 1}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e}{x - 1}\right) = 0$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right) = e$$
$$\lim_{x \to \infty}\left(\frac{e^{x} - e}{x - 1}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x} - e}{x - 1}\right) = 0$$
More at x→-oo
The graph