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x/sqrt(1+9*x^2)
Limit of the function/ x/sqrt(1+9*x^2)

Limit of the function x/sqrt(1+9*x^2)

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     /      x      \
 lim |-------------|
x->oo|   __________|
     |  /        2 |
     \\/  1 + 9*x  /
lim⁡x→∞(x9x2+1)\lim_{x \to \infty}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) 
Limit(x/sqrt(1 + 9*x^2), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞x=∞\lim_{x \to \infty} x = \infty
and limit for the denominator is
lim⁡x→∞9x2+1=∞\lim_{x \to \infty} \sqrt{9 x^{2} + 1} = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(x9x2+1)\lim_{x \to \infty}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right)
=
lim⁡x→∞(ddxxddx9x2+1)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sqrt{9 x^{2} + 1}}\right)
=
lim⁡x→∞(9x2+19x)\lim_{x \to \infty}\left(\frac{\sqrt{9 x^{2} + 1}}{9 x}\right)
=
lim⁡x→∞(9x2+19x)\lim_{x \to \infty}\left(\frac{\sqrt{9 x^{2} + 1}}{9 x}\right)
=
13\frac{1}{3}
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
02468-8-6-4-2-10100.5-0.5
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Rapid solution [src] 
1/3
13\frac{1}{3}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x9x2+1)=13\lim_{x \to \infty}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) = \frac{1}{3}
lim⁡x→0−(x9x2+1)=0\lim_{x \to 0^-}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) = 0
More at x→0 from the left
lim⁡x→0+(x9x2+1)=0\lim_{x \to 0^+}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) = 0
More at x→0 from the right
lim⁡x→1−(x9x2+1)=1010\lim_{x \to 1^-}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) = \frac{\sqrt{10}}{10}
More at x→1 from the left
lim⁡x→1+(x9x2+1)=1010\lim_{x \to 1^+}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) = \frac{\sqrt{10}}{10}
More at x→1 from the right
lim⁡x→−∞(x9x2+1)=−13\lim_{x \to -\infty}\left(\frac{x}{\sqrt{9 x^{2} + 1}}\right) = - \frac{1}{3}
More at x→-oo
The graph
Limit of the function x/sqrt(1+9*x^2)
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