Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-3*x^2+2*x+7*x^5)/(-5+x+x^5)
-
Limit of (4-x)/(4+x^2-5*x)
-
Limit of -6-3*x+4*x^2
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Limit of (3-x)^(x/(2-x))
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Identical expressions
- (three -x)^(x/(two -x))
- (3 minus x) to the power of (x divide by (2 minus x))
- (three minus x) to the power of (x divide by (two minus x))
- (3-x)(x/(2-x))
- 3-xx/2-x
- 3-x^x/2-x
- (3-x)^(x divide by (2-x))
-
Similar expressions
- (3-x)^(x/(2+x))
- (3+x)^(x/(2-x))
Limit of the function (3-x)^(x/(2-x))
The solution
You have entered
[src]
x
-----
2 - x
lim (3 - x)
x->2+
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}}$$
Limit((3 - x)^(x/(2 - x)), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}}$$
transform
do replacement
$$u = \frac{1}{2 - x}$$
then
$$\lim_{x \to 2^+} \left(1 + \frac{1}{\frac{1}{2 - x}}\right)^{\frac{x}{2 - x}}$$ =
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u \left(2 - \frac{1}{u}\right)}$$
=
$$\lim_{u \to 2^+}\left(\left(1 + \frac{1}{u}\right)^{\text{NaN}} \left(1 + \frac{1}{u}\right)^{\text{NaN}}\right)$$
=
$$\left(\lim_{u \to 2^+} \text{NaN}\right)^{2}$$
=
$$\lim_{u \to 2^+} \text{NaN}$$
=
$$\left(\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\text{NaN}}$$
The limit
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
The final answer:
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}} = e^{2}$$
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}}$$
transform
do replacement
$$u = \frac{1}{2 - x}$$
then
$$\lim_{x \to 2^+} \left(1 + \frac{1}{\frac{1}{2 - x}}\right)^{\frac{x}{2 - x}}$$ =
=
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u \left(2 - \frac{1}{u}\right)}$$
=
$$\lim_{u \to 2^+}\left(\left(1 + \frac{1}{u}\right)^{\text{NaN}} \left(1 + \frac{1}{u}\right)^{\text{NaN}}\right)$$
=
$$\left(\lim_{u \to 2^+} \text{NaN}\right)^{2}$$
=
$$\lim_{u \to 2^+} \text{NaN}$$
=
$$\left(\left(\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\text{NaN}}$$
The limit
$$\lim_{u \to 2^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
False
The final answer:
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}} = e^{2}$$
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]
x
-----
2 - x
lim (3 - x)
x->2+
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}}$$
2 e
$$e^{2}$$
= 7.38905609893065
x
-----
2 - x
lim (3 - x)
x->2-
$$\lim_{x \to 2^-} \left(3 - x\right)^{\frac{x}{2 - x}}$$
2 e
$$e^{2}$$
= 7.38905609893065
= 7.38905609893065
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 2^-} \left(3 - x\right)^{\frac{x}{2 - x}} = e^{2}$$
More at x→2 from the left
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}} = e^{2}$$
$$\lim_{x \to \infty} \left(3 - x\right)^{\frac{x}{2 - x}} = 0$$
More at x→oo
$$\lim_{x \to 0^-} \left(3 - x\right)^{\frac{x}{2 - x}} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(3 - x\right)^{\frac{x}{2 - x}} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(3 - x\right)^{\frac{x}{2 - x}} = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(3 - x\right)^{\frac{x}{2 - x}} = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(3 - x\right)^{\frac{x}{2 - x}} = 0$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+} \left(3 - x\right)^{\frac{x}{2 - x}} = e^{2}$$
$$\lim_{x \to \infty} \left(3 - x\right)^{\frac{x}{2 - x}} = 0$$
More at x→oo
$$\lim_{x \to 0^-} \left(3 - x\right)^{\frac{x}{2 - x}} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(3 - x\right)^{\frac{x}{2 - x}} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(3 - x\right)^{\frac{x}{2 - x}} = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(3 - x\right)^{\frac{x}{2 - x}} = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(3 - x\right)^{\frac{x}{2 - x}} = 0$$
More at x→-oo
The graph