Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-exp(-sin(2*x))+exp(tan(2*x)))/(-1+sin(x))
-
Limit of (e^(2*x)-e^(3*x))/(-x^2+atan(x))
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Limit of (-1+e^(2*x))/log(1-4*x)
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Limit of (-1+e^(x^2))/x
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Identical expressions
- (- one +e^(x^ two))/x
- ( minus 1 plus e to the power of (x squared )) divide by x
- ( minus one plus e to the power of (x to the power of two)) divide by x
- (-1+e(x2))/x
- -1+ex2/x
- (-1+e^(x²))/x
- (-1+e to the power of (x to the power of 2))/x
- -1+e^x^2/x
- (-1+e^(x^2)) divide by x
-
Similar expressions
- (1+e^(x^2))/x
- (-1-e^(x^2))/x
Limit of the function (-1+e^(x^2))/x
The solution
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[src]
/ / 2\\
| \x /|
|-1 + E |
lim |----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
Limit((-1 + E^(x^2))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(e^{x^{2}} - 1\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x^{2}} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(2 x e^{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(2 x\right)$$
=
$$\lim_{x \to 0^+}\left(2 x\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(e^{x^{2}} - 1\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x^{2}} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(2 x e^{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(2 x\right)$$
=
$$\lim_{x \to 0^+}\left(2 x\right)$$
=
$$0$$
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]
/ / 2\\
| \x /|
|-1 + E |
lim |----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
0
$$0$$
= 1.58521929052537e-30
/ / 2\\
| \x /|
|-1 + E |
lim |----------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{e^{x^{2}} - 1}{x}\right)$$
0
$$0$$
= -1.58521929052537e-30
= -1.58521929052537e-30
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{e^{x^{2}} - 1}{x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x^{2}} - 1}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{x^{2}} - 1}{x}\right) = -1 + e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x^{2}} - 1}{x}\right) = -1 + e$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x^{2}} - 1}{x}\right) = -\infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x^{2}} - 1}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{e^{x^{2}} - 1}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{x^{2}} - 1}{x}\right) = -1 + e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x^{2}} - 1}{x}\right) = -1 + e$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x^{2}} - 1}{x}\right) = -\infty$$
More at x→-oo
The graph