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Limit of the function/ x^2+x*y

Limit of the function x^2+x*y

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     / 2      \
 lim \x  + x*y/
x->oo
lim⁡x→∞(x2+xy)\lim_{x \to \infty}\left(x^{2} + x y\right) 
Limit(x^2 + x*y, x, oo, dir='-')
Detail solution
Let's take the limit
lim⁡x→∞(x2+xy)\lim_{x \to \infty}\left(x^{2} + x y\right)
Let's divide numerator and denominator by x^2:
lim⁡x→∞(x2+xy)\lim_{x \to \infty}\left(x^{2} + x y\right) =
lim⁡x→∞(1+yx1x2)\lim_{x \to \infty}\left(\frac{1 + \frac{y}{x}}{\frac{1}{x^{2}}}\right)
Do Replacement
u=1xu = \frac{1}{x}
then
lim⁡x→∞(1+yx1x2)=lim⁡u→0+(uy+1u2)\lim_{x \to \infty}\left(\frac{1 + \frac{y}{x}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u y + 1}{u^{2}}\right)
=
0y+10=∞\frac{0 y + 1}{0} = \infty

The final answer:
lim⁡x→∞(x2+xy)=∞\lim_{x \to \infty}\left(x^{2} + x y\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
Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x2+xy)=∞\lim_{x \to \infty}\left(x^{2} + x y\right) = \infty
lim⁡x→0−(x2+xy)=0\lim_{x \to 0^-}\left(x^{2} + x y\right) = 0
More at x→0 from the left
lim⁡x→0+(x2+xy)=0\lim_{x \to 0^+}\left(x^{2} + x y\right) = 0
More at x→0 from the right
lim⁡x→1−(x2+xy)=y+1\lim_{x \to 1^-}\left(x^{2} + x y\right) = y + 1
More at x→1 from the left
lim⁡x→1+(x2+xy)=y+1\lim_{x \to 1^+}\left(x^{2} + x y\right) = y + 1
More at x→1 from the right
lim⁡x→−∞(x2+xy)=∞\lim_{x \to -\infty}\left(x^{2} + x y\right) = \infty
More at x→-oo
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