Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of 2+((-1+x)/(3+x))^x
-
Limit of (-1+x)/(x+x^2)
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Limit of ((1+x)^4-(-1+x)^4)/((1+x)^4+(-1+x)^4)
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Limit of tan(6*x)/(3*x)
- Graphing y =:
- (1+x)^(1/x)
- Derivative of:
-
(1+x)^(1/x)
- Canonical form:
- (1+x)^(1/x)
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Identical expressions
- (one +x)^(one /x)
- (1 plus x) to the power of (1 divide by x)
- (one plus x) to the power of (one divide by x)
- (1+x)(1/x)
- 1+x1/x
- 1+x^1/x
- (1+x)^(1 divide by x)
-
Similar expressions
- (1-x)^(1/x)
Limit of the function (1+x)^(1/x)
The solution
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[src]
x _______ lim \/ 1 + x x->0+
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}}$$
Limit((1 + x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{1}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right) = e$$
The final answer:
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} = e$$
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{1}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right) = e$$
The final answer:
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} = e$$
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 _______ lim \/ 1 + x x->0+
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}}$$
E
$$e$$
= 2.71828182845905
x _______ lim \/ 1 + x x->0-
$$\lim_{x \to 0^-} \left(x + 1\right)^{\frac{1}{x}}$$
E
$$e$$
= 2.71828182845905
= 2.71828182845905
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(x + 1\right)^{\frac{1}{x}} = e$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} = e$$
$$\lim_{x \to \infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(x + 1\right)^{\frac{1}{x}} = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(x + 1\right)^{\frac{1}{x}} = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} = e$$
$$\lim_{x \to \infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(x + 1\right)^{\frac{1}{x}} = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(x + 1\right)^{\frac{1}{x}} = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
The graph