Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of 1-x-sin(x/2)/pi
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Limit of (1-cos(x)^2)/x
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Limit of (1-cos(4*x))/(4*x^2)
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Limit of (1+8*x)^(1/x)
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Identical expressions
- (one + eight *x)^(one /x)
- (1 plus 8 multiply by x) to the power of (1 divide by x)
- (one plus eight multiply by x) to the power of (one divide by x)
- (1+8*x)(1/x)
- 1+8*x1/x
- (1+8x)^(1/x)
- (1+8x)(1/x)
- 1+8x1/x
- 1+8x^1/x
- (1+8*x)^(1 divide by x)
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Similar expressions
- (1-8*x)^(1/x)
Limit of the function (1+8*x)^(1/x)
The solution
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[src]
x _________ lim \/ 1 + 8*x x->0+
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}}$$
Limit((1 + 8*x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{8 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{8}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8} = e^{8}$$
The final answer:
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}} = e^{8}$$
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{8 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{8}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{8 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{8} = e^{8}$$
The final answer:
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}} = e^{8}$$
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 + 8*x x->0+
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}}$$
8 e
$$e^{8}$$
= 2980.95798704173
x _________ lim \/ 1 + 8*x x->0-
$$\lim_{x \to 0^-} \left(8 x + 1\right)^{\frac{1}{x}}$$
8 e
$$e^{8}$$
exp(8)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(8 x + 1\right)^{\frac{1}{x}} = e^{8}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}} = e^{8}$$
$$\lim_{x \to \infty} \left(8 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(8 x + 1\right)^{\frac{1}{x}} = 9$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(8 x + 1\right)^{\frac{1}{x}} = 9$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(8 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(8 x + 1\right)^{\frac{1}{x}} = e^{8}$$
$$\lim_{x \to \infty} \left(8 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(8 x + 1\right)^{\frac{1}{x}} = 9$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(8 x + 1\right)^{\frac{1}{x}} = 9$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(8 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
The graph