Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-1+x)/(x+x^2)
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Limit of log(-5+x)/log(e^x-e^5)
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Limit of (-exp(-x)-2*x+exp(x))/(x-sin(x))
- Limit of e^(-n*x)/n
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Identical expressions
- (e^(four *x)-e^(three *x))/(-sin(three *x)+sin(four *x))
- (e to the power of (4 multiply by x) minus e to the power of (3 multiply by x)) divide by ( minus sinus of (3 multiply by x) plus sinus of (4 multiply by x))
- (e to the power of (four multiply by x) minus e to the power of (three multiply by x)) divide by ( minus sinus of (three multiply by x) plus sinus of (four multiply by x))
- (e(4*x)-e(3*x))/(-sin(3*x)+sin(4*x))
- e4*x-e3*x/-sin3*x+sin4*x
- (e^(4x)-e^(3x))/(-sin(3x)+sin(4x))
- (e(4x)-e(3x))/(-sin(3x)+sin(4x))
- e4x-e3x/-sin3x+sin4x
- e^4x-e^3x/-sin3x+sin4x
- (e^(4*x)-e^(3*x)) divide by (-sin(3*x)+sin(4*x))
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Similar expressions
- (e^(4*x)-e^(3*x))/(sin(3*x)+sin(4*x))
- (e^(4*x)-e^(3*x))/(-sin(3*x)-sin(4*x))
- (e^(4*x)+e^(3*x))/(-sin(3*x)+sin(4*x))
Limit of the function (e^(4*x)-e^(3*x))/(-sin(3*x)+sin(4*x))
The solution
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[src]
/ 4*x 3*x \
| E - E |
lim |--------------------|
x->0+\-sin(3*x) + sin(4*x)/
$$\lim_{x \to 0^+}\left(\frac{e^{4 x} - e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
Limit((E^(4*x) - E^(3*x))/(-sin(3*x) + sin(4*x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\left(e^{x} - 1\right) e^{3 x}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{e^{4 x} - e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\left(e^{x} - 1\right) e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x} - 1\right) e^{3 x}}{\frac{d}{d x} \left(- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 e^{4 x} - 3 e^{3 x}}{- 3 \cos{\left(3 x \right)} + 4 \cos{\left(4 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 e^{4 x} - 3 e^{3 x}}{- 3 \cos{\left(3 x \right)} + 4 \cos{\left(4 x \right)}}\right)$$
=
$$1$$
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
$$\lim_{x \to 0^+}\left(\left(e^{x} - 1\right) e^{3 x}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{e^{4 x} - e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\left(e^{x} - 1\right) e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x} - 1\right) e^{3 x}}{\frac{d}{d x} \left(- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 e^{4 x} - 3 e^{3 x}}{- 3 \cos{\left(3 x \right)} + 4 \cos{\left(4 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 e^{4 x} - 3 e^{3 x}}{- 3 \cos{\left(3 x \right)} + 4 \cos{\left(4 x \right)}}\right)$$
=
$$1$$
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
One‐sided limits
[src]
/ 4*x 3*x \
| E - E |
lim |--------------------|
x->0+\-sin(3*x) + sin(4*x)/
$$\lim_{x \to 0^+}\left(\frac{e^{4 x} - e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
1
$$1$$
= 1.0
/ 4*x 3*x \
| E - E |
lim |--------------------|
x->0-\-sin(3*x) + sin(4*x)/
$$\lim_{x \to 0^-}\left(\frac{e^{4 x} - e^{3 x}}{- \sin{\left(3 x \right)} + \sin{\left(4 x \right)}}\right)$$
1
$$1$$
= 1.0
= 1.0
The graph