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(x^2+6*x)/x
Limit of the function/ (x^2+6*x)/x

Limit of the function (x^2+6*x)/x

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     / 2      \
     |x  + 6*x|
 lim |--------|
x->0+\   x    /
lim⁡x→0+(x2+6xx)\lim_{x \to 0^+}\left(\frac{x^{2} + 6 x}{x}\right) 
Limit((x^2 + 6*x)/x, x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(x2+6xx)\lim_{x \to 0^+}\left(\frac{x^{2} + 6 x}{x}\right)
transform
lim⁡x→0+(x2+6xx)\lim_{x \to 0^+}\left(\frac{x^{2} + 6 x}{x}\right)
=
lim⁡x→0+(x(x+6)x)\lim_{x \to 0^+}\left(\frac{x \left(x + 6\right)}{x}\right)
=
lim⁡x→0+(x+6)=\lim_{x \to 0^+}\left(x + 6\right) =
6=6 = 
= 6

The final answer:
lim⁡x→0+(x2+6xx)=6\lim_{x \to 0^+}\left(\frac{x^{2} + 6 x}{x}\right) = 6
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] 
6
66
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(x2+6xx)=6\lim_{x \to 0^-}\left(\frac{x^{2} + 6 x}{x}\right) = 6
More at x→0 from the left
lim⁡x→0+(x2+6xx)=6\lim_{x \to 0^+}\left(\frac{x^{2} + 6 x}{x}\right) = 6
lim⁡x→∞(x2+6xx)=∞\lim_{x \to \infty}\left(\frac{x^{2} + 6 x}{x}\right) = \infty
More at x→oo
lim⁡x→1−(x2+6xx)=7\lim_{x \to 1^-}\left(\frac{x^{2} + 6 x}{x}\right) = 7
More at x→1 from the left
lim⁡x→1+(x2+6xx)=7\lim_{x \to 1^+}\left(\frac{x^{2} + 6 x}{x}\right) = 7
More at x→1 from the right
lim⁡x→−∞(x2+6xx)=−∞\lim_{x \to -\infty}\left(\frac{x^{2} + 6 x}{x}\right) = -\infty
More at x→-oo
One‐sided limits [src] 
     / 2      \
     |x  + 6*x|
 lim |--------|
x->0+\   x    /
lim⁡x→0+(x2+6xx)\lim_{x \to 0^+}\left(\frac{x^{2} + 6 x}{x}\right) 
6
66 
= 6.0
     / 2      \
     |x  + 6*x|
 lim |--------|
x->0-\   x    /
lim⁡x→0−(x2+6xx)\lim_{x \to 0^-}\left(\frac{x^{2} + 6 x}{x}\right) 
6
66 
= 6.0
= 6.0
Numerical answer [src] 
6.0
6.0
The graph
Limit of the function (x^2+6*x)/x
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