Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (3-3*x^2+4*x^4+6*x^3)/(2*x^2+7*x^4)
-
Limit of ((5+4*x)/(-1+5*x))^(1+3*x)
-
Limit of (1+x^3-2*x)/(8+x^10-9*x)
- Limit of x^2+a/(x^3-a^3)-x*(1+a)
-
Identical expressions
- (two -x)/(- four +x^ two)
- (2 minus x) divide by ( minus 4 plus x squared )
- (two minus 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)
Limit of the function (2-x)/(-4+x^2)
The solution
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[src]
/ 2 - x \
lim |-------|
x->2+| 2|
\-4 + x /
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
Limit((2 - x)/(-4 + x^2), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{2 - x}{\left(x - 2\right) \left(x + 2\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(- \frac{1}{x + 2}\right) = $$
$$- \frac{1}{2 + 2} = $$
The final answer:
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{4}$$
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{2 - x}{\left(x - 2\right) \left(x + 2\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(- \frac{1}{x + 2}\right) = $$
$$- \frac{1}{2 + 2} = $$
= -1/4
The final answer:
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{4}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 2^+}\left(2 - x\right) = 0$$
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.
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(2 - x\right)}{\frac{d}{d x} \left(x^{2} - 4\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(- \frac{1}{2 x}\right)$$
=
$$\lim_{x \to 2^+} - \frac{1}{4}$$
=
$$\lim_{x \to 2^+} - \frac{1}{4}$$
=
$$- \frac{1}{4}$$
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
$$\lim_{x \to 2^+}\left(2 - x\right) = 0$$
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.
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(2 - x\right)}{\frac{d}{d x} \left(x^{2} - 4\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(- \frac{1}{2 x}\right)$$
=
$$\lim_{x \to 2^+} - \frac{1}{4}$$
=
$$\lim_{x \to 2^+} - \frac{1}{4}$$
=
$$- \frac{1}{4}$$
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 /
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right)$$
-1/4
$$- \frac{1}{4}$$
= -0.25
/ 2 - x \
lim |-------|
x->2-| 2|
\-4 + x /
$$\lim_{x \to 2^-}\left(\frac{2 - x}{x^{2} - 4}\right)$$
-1/4
$$- \frac{1}{4}$$
= -0.25
= -0.25
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 2^-}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{4}$$
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{4}$$
$$\lim_{x \to \infty}\left(\frac{2 - x}{x^{2} - 4}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 - x}{x^{2} - 4}\right) = 0$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{4}$$
$$\lim_{x \to \infty}\left(\frac{2 - x}{x^{2} - 4}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 - x}{x^{2} - 4}\right) = - \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 - x}{x^{2} - 4}\right) = 0$$
More at x→-oo
The graph