Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-3+x^2-2*x)/(-3+x)
-
Limit of 5^x-cos(x)
-
Limit of (x/2)^(1/(-2+x))
-
Limit of (1-3/x)^x
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Identical expressions
- (x/ two)^(one /(- two +x))
- (x divide by 2) to the power of (1 divide by ( minus 2 plus x))
- (x divide by two) to the power of (one divide by ( minus two plus x))
- (x/2)(1/(-2+x))
- x/21/-2+x
- x/2^1/-2+x
- (x divide by 2)^(1 divide by (-2+x))
-
Similar expressions
- (x/2)^(1/(-2-x))
- (x/2)^(1/(2+x))
Limit of the function (x/2)^(1/(-2+x))
The solution
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[src]
1
1*------
-2 + x
/x\
lim |-|
x->2+\2/
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}}$$
Limit((x/2)^(1/(-2 + x)), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}}$$
transform
do replacement
$$u = \frac{1}{1 \left(\frac{x}{2} - 1\right)}$$
then
$$\lim_{x \to 2^+} \left(1 + \frac{1}{\frac{1}{\frac{x}{2} - 1}}\right)^{\frac{1}{x - 2}}$$ =
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{u}{2}}$$
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{u}{2}}$$
=
$$\sqrt{\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)}$$
The limit
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\sqrt{\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)} = e^{\frac{1}{2}}$$
The final answer:
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = e^{\frac{1}{2}}$$
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}}$$
transform
do replacement
$$u = \frac{1}{1 \left(\frac{x}{2} - 1\right)}$$
then
$$\lim_{x \to 2^+} \left(1 + \frac{1}{\frac{1}{\frac{x}{2} - 1}}\right)^{\frac{1}{x - 2}}$$ =
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{u}{2}}$$
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{\frac{u}{2}}$$
=
$$\sqrt{\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)}$$
The limit
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\sqrt{\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)} = e^{\frac{1}{2}}$$
The final answer:
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = e^{\frac{1}{2}}$$
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]
1
1*------
-2 + x
/x\
lim |-|
x->2+\2/
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}}$$
1/2 e
$$e^{\frac{1}{2}}$$
= 1.64872127070013
1
1*------
-2 + x
/x\
lim |-|
x->2-\2/
$$\lim_{x \to 2^-} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}}$$
1/2 e
$$e^{\frac{1}{2}}$$
= 1.64872127070013
= 1.64872127070013
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 2^-} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = e^{\frac{1}{2}}$$
More at x→2 from the left
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = e^{\frac{1}{2}}$$
$$\lim_{x \to \infty} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 1$$
More at x→oo
$$\lim_{x \to 0^-} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = - \infty i$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 1$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = e^{\frac{1}{2}}$$
$$\lim_{x \to \infty} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 1$$
More at x→oo
$$\lim_{x \to 0^-} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = - \infty i$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(\frac{x}{2}\right)^{1 \cdot \frac{1}{x - 2}} = 1$$
More at x→-oo
The graph