Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of ((3+x)^2+(3-x)^2)/((3-x)^2-(3+x)^2)
-
Limit of (-1+x)/(-2+sqrt(3+x))
-
Limit of (-x+tan(x))/x^3
-
Limit of (-1+sqrt(x))/(-3+x)
- Graphing y =:
- x^2/(4-x^2)
- Integral of d{x}:
-
x^2/(4-x^2)
-
Identical expressions
- x^ two /(four -x^ two)
- x squared divide by (4 minus x squared )
- x to the power of two divide by (four minus x to the power of two)
- x2/(4-x2)
- x2/4-x2
- x²/(4-x²)
- x to the power of 2/(4-x to the power of 2)
- x^2/4-x^2
- x^2 divide by (4-x^2)
-
Similar expressions
- (8+x^3-4*x^2)/(4-x^2)
- ((1-x^2)/(4-x^2))^(4*x^2)
- x^2/(4+x^2)
- (-6-x+5*x^2)/(4-x^2+3*x)
Limit of the function x^2/(4-x^2)
The solution
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/ 2 \
| x |
lim |------|
x->oo| 2|
\4 - x /
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$
Limit(x^2/(4 - x^2), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{-1 + \frac{4}{x^{2}}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{-1 + \frac{4}{x^{2}}} = \lim_{u \to 0^+} \frac{1}{4 u^{2} - 1}$$
=
$$\frac{1}{-1 + 4 \cdot 0^{2}} = -1$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{-1 + \frac{4}{x^{2}}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{-1 + \frac{4}{x^{2}}} = \lim_{u \to 0^+} \frac{1}{4 u^{2} - 1}$$
=
$$\frac{1}{-1 + 4 \cdot 0^{2}} = -1$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} x^{2} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(4 - x^{2}\right) = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} \left(4 - x^{2}\right)}\right)$$
=
$$\lim_{x \to \infty} -1$$
=
$$\lim_{x \to \infty} -1$$
=
$$-1$$
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
$$\lim_{x \to \infty} x^{2} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(4 - x^{2}\right) = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{2}}{\frac{d}{d x} \left(4 - x^{2}\right)}\right)$$
=
$$\lim_{x \to \infty} -1$$
=
$$\lim_{x \to \infty} -1$$
=
$$-1$$
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
$$\lim_{x \to \infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1$$
$$\lim_{x \to 0^-}\left(\frac{x^{2}}{4 - x^{2}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{2}}{4 - x^{2}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x^{2}}{4 - x^{2}}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{2}}{4 - x^{2}}\right) = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x^{2}}{4 - x^{2}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{2}}{4 - x^{2}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x^{2}}{4 - x^{2}}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{2}}{4 - x^{2}}\right) = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{2}}{4 - x^{2}}\right) = -1$$
More at x→-oo
The graph