Mister Exam
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How to use it?
- Limit of the function:
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Limit of (-1+e^(3*x))/x
-
Limit of sin(8*x)/x
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Limit of cot(3*x)/cot(5*x)
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Limit of (sin(x)+sin(3*x))/(10*x)
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Identical expressions
- (- one +e^(three *x))/x
- ( minus 1 plus e to the power of (3 multiply by x)) divide by x
- ( minus one plus e to the power of (three multiply by x)) divide by x
- (-1+e(3*x))/x
- -1+e3*x/x
- (-1+e^(3x))/x
- (-1+e(3x))/x
- -1+e3x/x
- -1+e^3x/x
- (-1+e^(3*x)) divide by x
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Similar expressions
- (-1-e^(3*x))/x
- (1+e^(3*x))/x
Limit of the function (-1+e^(3*x))/x
The solution
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[src]
/ 3*x\
|-1 + E |
lim |---------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)$$
Limit((-1 + E^(3*x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(e^{3 x} - 1\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{3 x} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(3 e^{3 x}\right)$$
=
$$\lim_{x \to 0^+} 3$$
=
$$\lim_{x \to 0^+} 3$$
=
$$3$$
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
$$\lim_{x \to 0^+}\left(e^{3 x} - 1\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{3 x} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(3 e^{3 x}\right)$$
=
$$\lim_{x \to 0^+} 3$$
=
$$\lim_{x \to 0^+} 3$$
=
$$3$$
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]
/ 3*x\
|-1 + E |
lim |---------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)$$
3
$$3$$
= 3.0
/ 3*x\
|-1 + E |
lim |---------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{e^{3 x} - 1}{x}\right)$$
3
$$3$$
= 3.0
= 3.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{e^{3 x} - 1}{x}\right) = 3$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{e^{3 x} - 1}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{3 x} - 1}{x}\right) = -1 + e^{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{3 x} - 1}{x}\right) = -1 + e^{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{3 x} - 1}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right) = 3$$
$$\lim_{x \to \infty}\left(\frac{e^{3 x} - 1}{x}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{3 x} - 1}{x}\right) = -1 + e^{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{3 x} - 1}{x}\right) = -1 + e^{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{3 x} - 1}{x}\right) = 0$$
More at x→-oo
The graph