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Limit of the function/ x+2/x

Limit of the function x+2/x

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     /    2\
 lim |x + -|
x->oo\    x/
lim⁡x→∞(x+2x)\lim_{x \to \infty}\left(x + \frac{2}{x}\right) 
Limit(x + 2/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞(x2+2)=∞\lim_{x \to \infty}\left(x^{2} + 2\right) = \infty
and limit for the denominator is
lim⁡x→∞x=∞\lim_{x \to \infty} x = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(x+2x)\lim_{x \to \infty}\left(x + \frac{2}{x}\right)
=
Let's transform the function under the limit a few
lim⁡x→∞(x2+2x)\lim_{x \to \infty}\left(\frac{x^{2} + 2}{x}\right)
=
lim⁡x→∞(ddx(x2+2)ddxx)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} + 2\right)}{\frac{d}{d x} x}\right)
=
lim⁡x→∞(2x)\lim_{x \to \infty}\left(2 x\right)
=
lim⁡x→∞(2x)\lim_{x \to \infty}\left(2 x\right)
=
∞\infty
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
02468-8-6-4-2-1010-5050
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x+2x)=∞\lim_{x \to \infty}\left(x + \frac{2}{x}\right) = \infty
lim⁡x→0−(x+2x)=−∞\lim_{x \to 0^-}\left(x + \frac{2}{x}\right) = -\infty
More at x→0 from the left
lim⁡x→0+(x+2x)=∞\lim_{x \to 0^+}\left(x + \frac{2}{x}\right) = \infty
More at x→0 from the right
lim⁡x→1−(x+2x)=3\lim_{x \to 1^-}\left(x + \frac{2}{x}\right) = 3
More at x→1 from the left
lim⁡x→1+(x+2x)=3\lim_{x \to 1^+}\left(x + \frac{2}{x}\right) = 3
More at x→1 from the right
lim⁡x→−∞(x+2x)=−∞\lim_{x \to -\infty}\left(x + \frac{2}{x}\right) = -\infty
More at x→-oo
Rapid solution [src] 
oo
∞\infty
Expand and simplify
The graph
Limit of the function x+2/x
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