Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-3*e^(4*x)-2*e^(-x)+5*e^(2*x))/(-4*sqrt(1+5*x)+4*cos(3*x)+5*sin(2*x))
-
Limit of (x/2)^(5/(-2+x))
-
Limit of (-4+x^2)/(6+x^2+5*x)
-
Limit of ((-1+x)/(1+x))^(1+x)
-
Identical expressions
- (x/ two)^(five /(- two +x))
- (x divide by 2) to the power of (5 divide by ( minus 2 plus x))
- (x divide by two) to the power of (five divide by ( minus two plus x))
- (x/2)(5/(-2+x))
- x/25/-2+x
- x/2^5/-2+x
- (x divide by 2)^(5 divide by (-2+x))
-
Similar expressions
- (x/2)^(5/(2+x))
- (x/2)^(5/(-2-x))
Limit of the function (x/2)^(5/(-2+x))
The solution
You have entered
[src]
5
------
-2 + x
/x\
lim |-|
x->2+\2/
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}}$$
Limit((x/2)^(5/(-2 + x)), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}}$$
transform
do replacement
$$u = \frac{1}{\frac{x}{2} - 1}$$
then
$$\lim_{x \to 2^+} \left(1 + \frac{1}{\frac{1}{\frac{x}{2} - 1}}\right)^{\frac{5}{x - 2}}$$ =
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{2}}$$
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{2}}$$
=
$$\left(\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{2}}$$
The limit
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{2}} = e^{\frac{5}{2}}$$
The final answer:
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = e^{\frac{5}{2}}$$
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}}$$
transform
do replacement
$$u = \frac{1}{\frac{x}{2} - 1}$$
then
$$\lim_{x \to 2^+} \left(1 + \frac{1}{\frac{1}{\frac{x}{2} - 1}}\right)^{\frac{5}{x - 2}}$$ =
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{2}}$$
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{2}}$$
=
$$\left(\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{2}}$$
The limit
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{2}} = e^{\frac{5}{2}}$$
The final answer:
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = e^{\frac{5}{2}}$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
One‐sided limits
[src]
5
------
-2 + x
/x\
lim |-|
x->2+\2/
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}}$$
5/2 e
$$e^{\frac{5}{2}}$$
= 12.1824939607035
5
------
-2 + x
/x\
lim |-|
x->2-\2/
$$\lim_{x \to 2^-} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}}$$
5/2 e
$$e^{\frac{5}{2}}$$
= 12.1824939607035
= 12.1824939607035
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 2^-} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = e^{\frac{5}{2}}$$
More at x→2 from the left
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = e^{\frac{5}{2}}$$
$$\lim_{x \to \infty} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 1$$
More at x→oo
$$\lim_{x \to 0^-} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = - \infty i$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 32$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 32$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 1$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = e^{\frac{5}{2}}$$
$$\lim_{x \to \infty} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 1$$
More at x→oo
$$\lim_{x \to 0^-} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = - \infty i$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 32$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 32$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(\frac{x}{2}\right)^{\frac{5}{x - 2}} = 1$$
More at x→-oo
The graph