Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
- Limit of 2016+965*e^(-168*x)*sin(-115+547*x)
-
Limit of (19-32*x+9*x^4+16*x^6)/(38-12*x^5+8*x^6+12*x)
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Limit of 7+7*x/3
- Limit of (-1+x^m)/(-1+x)
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Identical expressions
- (two *x^ two + two *x^ three)/(- three *x^ two + five *x^ three)
- (2 multiply by x squared plus 2 multiply by x cubed ) divide by ( minus 3 multiply by x squared plus 5 multiply by x cubed )
- (two multiply by x to the power of two plus two multiply by x to the power of three) divide by ( minus three multiply by x to the power of two plus five multiply by x to the power of three)
- (2*x2+2*x3)/(-3*x2+5*x3)
- 2*x2+2*x3/-3*x2+5*x3
- (2*x²+2*x³)/(-3*x²+5*x³)
- (2*x to the power of 2+2*x to the power of 3)/(-3*x to the power of 2+5*x to the power of 3)
- (2x^2+2x^3)/(-3x^2+5x^3)
- (2x2+2x3)/(-3x2+5x3)
- 2x2+2x3/-3x2+5x3
- 2x^2+2x^3/-3x^2+5x^3
- (2*x^2+2*x^3) divide by (-3*x^2+5*x^3)
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Similar expressions
- (2*x^2-2*x^3)/(-3*x^2+5*x^3)
- (2*x^2+2*x^3)/(-3*x^2-5*x^3)
- (2*x^2+2*x^3)/(3*x^2+5*x^3)
Limit of the function (2*x^2+2*x^3)/(-3*x^2+5*x^3)
The solution
You have entered
[src]
/ 2 3 \
| 2*x + 2*x |
lim |-------------|
x->0+| 2 3|
\- 3*x + 5*x /
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
Limit((2*x^2 + 2*x^3)/(-3*x^2 + 5*x^3), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x^{2} \left(x + 1\right)}{x^{2} \left(5 x - 3\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(x + 1\right)}{5 x - 3}\right) = $$
$$\frac{2}{-3 + 0 \cdot 5} = $$
The final answer:
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = - \frac{2}{3}$$
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 x^{2} \left(x + 1\right)}{x^{2} \left(5 x - 3\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \left(x + 1\right)}{5 x - 3}\right) = $$
$$\frac{2}{-3 + 0 \cdot 5} = $$
= -2/3
The final answer:
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = - \frac{2}{3}$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
One‐sided limits
[src]
/ 2 3 \
| 2*x + 2*x |
lim |-------------|
x->0+| 2 3|
\- 3*x + 5*x /
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
-2/3
$$- \frac{2}{3}$$
= -0.666666666666667
/ 2 3 \
| 2*x + 2*x |
lim |-------------|
x->0-| 2 3|
\- 3*x + 5*x /
$$\lim_{x \to 0^-}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right)$$
-2/3
$$- \frac{2}{3}$$
= -0.666666666666667
= -0.666666666666667
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = - \frac{2}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = - \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = \frac{2}{5}$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = \frac{2}{5}$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = - \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = \frac{2}{5}$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x^{3} + 2 x^{2}}{5 x^{3} - 3 x^{2}}\right) = \frac{2}{5}$$
More at x→-oo
The graph