Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (4-x^2)/(3-x^2)
-
Limit of ((1+2*x)/(2+2*x))^x
-
Limit of (1-cos(2*x))*sin(x)/x
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Limit of (1+2*x)^(2/x)
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Identical expressions
- (one + two *x)^(two /x)
- (1 plus 2 multiply by x) to the power of (2 divide by x)
- (one plus two multiply by x) to the power of (two divide by x)
- (1+2*x)(2/x)
- 1+2*x2/x
- (1+2x)^(2/x)
- (1+2x)(2/x)
- 1+2x2/x
- 1+2x^2/x
- (1+2*x)^(2 divide by x)
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Similar expressions
- (1-2*x)^(2/x)
- (-1+2*x^2)/(x^2-2*x)
Limit of the function (1+2*x)^(2/x)
The solution
You have entered
[src]
2
-
x
lim (1 + 2*x)
x->0+
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}$$
Limit((1 + 2*x)^(2/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}$$
transform
do replacement
$$u = \frac{1}{2 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{2}{\frac{1}{x}}\right)^{\frac{2}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4} = e^{4}$$
The final answer:
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}$$
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}$$
transform
do replacement
$$u = \frac{1}{2 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{2}{\frac{1}{x}}\right)^{\frac{2}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4} = e^{4}$$
The final answer:
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}$$
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]
2
-
x
lim (1 + 2*x)
x->0+
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}$$
4 e
$$e^{4}$$
= 54.5981500331442
2
-
x
lim (1 + 2*x)
x->0-
$$\lim_{x \to 0^-} \left(2 x + 1\right)^{\frac{2}{x}}$$
4 e
$$e^{4}$$
exp(4)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}$$
$$\lim_{x \to \infty} \left(2 x + 1\right)^{\frac{2}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(2 x + 1\right)^{\frac{2}{x}} = 9$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(2 x + 1\right)^{\frac{2}{x}} = 9$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(2 x + 1\right)^{\frac{2}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}$$
$$\lim_{x \to \infty} \left(2 x + 1\right)^{\frac{2}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(2 x + 1\right)^{\frac{2}{x}} = 9$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(2 x + 1\right)^{\frac{2}{x}} = 9$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(2 x + 1\right)^{\frac{2}{x}} = 1$$
More at x→-oo
The graph