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x^3-x^4
Limit of the function/ x^3-x^4

Limit of the function x^3-x^4

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     / 3    4\
 lim \x  - x /
x->oo
lim⁡x→∞(−x4+x3)\lim_{x \to \infty}\left(- x^{4} + x^{3}\right) 
Limit(x^3 - x^4, x, oo, dir='-')
Detail solution
Let's take the limit
lim⁡x→∞(−x4+x3)\lim_{x \to \infty}\left(- x^{4} + x^{3}\right)
Let's divide numerator and denominator by x^4:
lim⁡x→∞(−x4+x3)\lim_{x \to \infty}\left(- x^{4} + x^{3}\right) =
lim⁡x→∞(−1+1x1x4)\lim_{x \to \infty}\left(\frac{-1 + \frac{1}{x}}{\frac{1}{x^{4}}}\right)
Do Replacement
u=1xu = \frac{1}{x}
then
lim⁡x→∞(−1+1x1x4)=lim⁡u→0+(u−1u4)\lim_{x \to \infty}\left(\frac{-1 + \frac{1}{x}}{\frac{1}{x^{4}}}\right) = \lim_{u \to 0^+}\left(\frac{u - 1}{u^{4}}\right)
=
−10=−∞\frac{-1}{0} = -\infty

The final answer:
lim⁡x→∞(−x4+x3)=−∞\lim_{x \to \infty}\left(- x^{4} + x^{3}\right) = -\infty
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
02468-8-6-4-2-1010-2000010000
Plot the graph
Rapid solution [src] 
-oo
−∞-\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(−x4+x3)=−∞\lim_{x \to \infty}\left(- x^{4} + x^{3}\right) = -\infty
lim⁡x→0−(−x4+x3)=0\lim_{x \to 0^-}\left(- x^{4} + x^{3}\right) = 0
More at x→0 from the left
lim⁡x→0+(−x4+x3)=0\lim_{x \to 0^+}\left(- x^{4} + x^{3}\right) = 0
More at x→0 from the right
lim⁡x→1−(−x4+x3)=0\lim_{x \to 1^-}\left(- x^{4} + x^{3}\right) = 0
More at x→1 from the left
lim⁡x→1+(−x4+x3)=0\lim_{x \to 1^+}\left(- x^{4} + x^{3}\right) = 0
More at x→1 from the right
lim⁡x→−∞(−x4+x3)=−∞\lim_{x \to -\infty}\left(- x^{4} + x^{3}\right) = -\infty
More at x→-oo
The graph
Limit of the function x^3-x^4
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