Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (1+x)/(2+(-7+x)^(1/3))
-
Limit of (-1+x)*cos(1/(-1+x))
-
Limit of x/(1-(1+x)^(1/3))
-
Limit of tan(7*x)^2/sin(4*x^2)
- Graphing y =:
- (-2+x)/(-4+x^2)
-
Identical expressions
- (- two +x)/(- four +x^ two)
- ( minus 2 plus x) divide by ( minus 4 plus x squared )
- ( minus two plus x) divide by ( minus four plus x to the power of two)
- (-2+x)/(-4+x2)
- -2+x/-4+x2
- (-2+x)/(-4+x²)
- (-2+x)/(-4+x to the power of 2)
- -2+x/-4+x^2
- (-2+x) divide by (-4+x^2)
-
Similar expressions
- (-2-x)/(-4+x^2)
- (-2+x)/(4+x^2)
- (-2+x)/(-4-x^2)
- (2+x)/(-4+x^2)
Limit of the function (-2+x)/(-4+x^2)
The solution
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[src]
/ -2 + x\
lim |-------|
x->2+| 2|
\-4 + x /
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
Limit((-2 + x)/(-4 + x^2), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{x - 2}{\left(x - 2\right) \left(x + 2\right)}\right)$$
=
$$\lim_{x \to 2^+} \frac{1}{x + 2} = $$
$$\frac{1}{2 + 2} = $$
The final answer:
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{4}$$
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{x - 2}{\left(x - 2\right) \left(x + 2\right)}\right)$$
=
$$\lim_{x \to 2^+} \frac{1}{x + 2} = $$
$$\frac{1}{2 + 2} = $$
= 1/4
The final answer:
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{4}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 2^+}\left(x - 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 2^+}\left(x^{2} - 4\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(x - 2\right)}{\frac{d}{d x} \left(x^{2} - 4\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{1}{2 x}\right)$$
=
$$\lim_{x \to 2^+} \frac{1}{4}$$
=
$$\lim_{x \to 2^+} \frac{1}{4}$$
=
$$\frac{1}{4}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 2^+}\left(x - 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 2^+}\left(x^{2} - 4\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(x - 2\right)}{\frac{d}{d x} \left(x^{2} - 4\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{1}{2 x}\right)$$
=
$$\lim_{x \to 2^+} \frac{1}{4}$$
=
$$\lim_{x \to 2^+} \frac{1}{4}$$
=
$$\frac{1}{4}$$
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
One‐sided limits
[src]
/ -2 + x\
lim |-------|
x->2+| 2|
\-4 + x /
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right)$$
1/4
$$\frac{1}{4}$$
= 0.25
/ -2 + x\
lim |-------|
x->2-| 2|
\-4 + x /
$$\lim_{x \to 2^-}\left(\frac{x - 2}{x^{2} - 4}\right)$$
1/4
$$\frac{1}{4}$$
= 0.25
= 0.25
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 2^-}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{4}$$
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{4}$$
$$\lim_{x \to \infty}\left(\frac{x - 2}{x^{2} - 4}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 2}{x^{2} - 4}\right) = 0$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{4}$$
$$\lim_{x \to \infty}\left(\frac{x - 2}{x^{2} - 4}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 2}{x^{2} - 4}\right) = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 2}{x^{2} - 4}\right) = 0$$
More at x→-oo
The graph