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(x^2-5*x)/x
Limit of the function/ (x^2-5*x)/x

Limit of the function (x^2-5*x)/x

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The solution

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     / 2      \
     |x  - 5*x|
 lim |--------|
x->0+\   x    /
lim⁡x→0+(x2−5xx)\lim_{x \to 0^+}\left(\frac{x^{2} - 5 x}{x}\right) 
Limit((x^2 - 5*x)/x, x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(x2−5xx)\lim_{x \to 0^+}\left(\frac{x^{2} - 5 x}{x}\right)
transform
lim⁡x→0+(x2−5xx)\lim_{x \to 0^+}\left(\frac{x^{2} - 5 x}{x}\right)
=
lim⁡x→0+(x(x−5)x)\lim_{x \to 0^+}\left(\frac{x \left(x - 5\right)}{x}\right)
=
lim⁡x→0+(x−5)=\lim_{x \to 0^+}\left(x - 5\right) =
−5=-5 = 
= -5

The final answer:
lim⁡x→0+(x2−5xx)=−5\lim_{x \to 0^+}\left(\frac{x^{2} - 5 x}{x}\right) = -5
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
02468-8-6-4-2-1010-2020
Plot the graph
Rapid solution [src] 
-5
−5-5
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(x2−5xx)=−5\lim_{x \to 0^-}\left(\frac{x^{2} - 5 x}{x}\right) = -5
More at x→0 from the left
lim⁡x→0+(x2−5xx)=−5\lim_{x \to 0^+}\left(\frac{x^{2} - 5 x}{x}\right) = -5
lim⁡x→∞(x2−5xx)=∞\lim_{x \to \infty}\left(\frac{x^{2} - 5 x}{x}\right) = \infty
More at x→oo
lim⁡x→1−(x2−5xx)=−4\lim_{x \to 1^-}\left(\frac{x^{2} - 5 x}{x}\right) = -4
More at x→1 from the left
lim⁡x→1+(x2−5xx)=−4\lim_{x \to 1^+}\left(\frac{x^{2} - 5 x}{x}\right) = -4
More at x→1 from the right
lim⁡x→−∞(x2−5xx)=−∞\lim_{x \to -\infty}\left(\frac{x^{2} - 5 x}{x}\right) = -\infty
More at x→-oo
One‐sided limits [src] 
     / 2      \
     |x  - 5*x|
 lim |--------|
x->0+\   x    /
lim⁡x→0+(x2−5xx)\lim_{x \to 0^+}\left(\frac{x^{2} - 5 x}{x}\right) 
-5
−5-5 
= -5.0
     / 2      \
     |x  - 5*x|
 lim |--------|
x->0-\   x    /
lim⁡x→0−(x2−5xx)\lim_{x \to 0^-}\left(\frac{x^{2} - 5 x}{x}\right) 
-5
−5-5 
= -5.0
= -5.0
Numerical answer [src] 
-5.0
-5.0
The graph
Limit of the function (x^2-5*x)/x
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