Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of -2+x^3+6*x
-
Limit of ((-2+x)/x)^(2*x)
-
Limit of (-2+x)*log(-2+x)
-
Limit of -6+x^2+5*x
- Integral of d{x}:
-
x/(9+x^2)
- Graphing y =:
- x/(9+x^2)
-
Identical expressions
- x/(nine +x^ two)
- x divide by (9 plus x squared )
- x divide by (nine plus x to the power of two)
- x/(9+x2)
- x/9+x2
- x/(9+x²)
- x/(9+x to the power of 2)
- x/9+x^2
- x divide by (9+x^2)
-
Similar expressions
- x/(9-x^2)
Limit of the function x/(9+x^2)
The solution
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/ x \
lim |------|
x->-oo| 2|
\9 + x /
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$
Limit(x/(9 + x^2), x, -oo)
Detail solution
Let's take the limit
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$ =
$$\lim_{x \to -\infty}\left(\frac{1}{x \left(1 + \frac{9}{x^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to -\infty}\left(\frac{1}{x \left(1 + \frac{9}{x^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u}{9 u^{2} + 1}\right)$$
=
$$\frac{0}{9 \cdot 0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right) = 0$$
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$ =
$$\lim_{x \to -\infty}\left(\frac{1}{x \left(1 + \frac{9}{x^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to -\infty}\left(\frac{1}{x \left(1 + \frac{9}{x^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u}{9 u^{2} + 1}\right)$$
=
$$\frac{0}{9 \cdot 0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right) = 0$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to -\infty} x = -\infty$$
and limit for the denominator is
$$\lim_{x \to -\infty}\left(x^{2} + 9\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(x^{2} + 9\right)}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{1}{2 x}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{1}{2 x}\right)$$
=
$$0$$
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)
-oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to -\infty} x = -\infty$$
and limit for the denominator is
$$\lim_{x \to -\infty}\left(x^{2} + 9\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(x^{2} + 9\right)}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{1}{2 x}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{1}{2 x}\right)$$
=
$$0$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to -\infty}\left(\frac{x}{x^{2} + 9}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{x^{2} + 9}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x}{x^{2} + 9}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{x^{2} + 9}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{x^{2} + 9}\right) = \frac{1}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{x^{2} + 9}\right) = \frac{1}{10}$$
More at x→1 from the right
$$\lim_{x \to \infty}\left(\frac{x}{x^{2} + 9}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x}{x^{2} + 9}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{x^{2} + 9}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{x^{2} + 9}\right) = \frac{1}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{x^{2} + 9}\right) = \frac{1}{10}$$
More at x→1 from the right
The graph