Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (1-4*x)^(1/x)
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Limit of (-16+x^2+6*x)/(-2-5*x+3*x^2)
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Limit of (1+x)^(2/3)-(-1+x)^(2/3)
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Limit of 1/3+x/3
- Graphing y =:
- (1-4*x)^(1/x)
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Identical expressions
- (one - four *x)^(one /x)
- (1 minus 4 multiply by x) to the power of (1 divide by x)
- (one minus four multiply by x) to the power of (one divide by x)
- (1-4*x)(1/x)
- 1-4*x1/x
- (1-4x)^(1/x)
- (1-4x)(1/x)
- 1-4x1/x
- 1-4x^1/x
- (1-4*x)^(1 divide by x)
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Similar expressions
- (1+4*x)^(1/x)
Limit of the function (1-4*x)^(1/x)
The solution
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[src]
x _________ lim \/ 1 - 4*x x->0+
$$\lim_{x \to 0^+} \left(1 - 4 x\right)^{\frac{1}{x}}$$
Limit((1 - 4*x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(1 - 4 x\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{\left(-4\right) x}$$
then
$$\lim_{x \to 0^+} \left(1 - \frac{4}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 4 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 4 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-4}$$
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:
$$\lim_{x \to 0^+} \left(1 - 4 x\right)^{\frac{1}{x}} = e^{-4}$$
$$\lim_{x \to 0^+} \left(1 - 4 x\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{\left(-4\right) x}$$
then
$$\lim_{x \to 0^+} \left(1 - \frac{4}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 4 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 4 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-4}$$
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:
$$\lim_{x \to 0^+} \left(1 - 4 x\right)^{\frac{1}{x}} = e^{-4}$$
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]
x _________ lim \/ 1 - 4*x x->0+
$$\lim_{x \to 0^+} \left(1 - 4 x\right)^{\frac{1}{x}}$$
-4 e
$$e^{-4}$$
= 0.0183156388887342
x _________ lim \/ 1 - 4*x x->0-
$$\lim_{x \to 0^-} \left(1 - 4 x\right)^{\frac{1}{x}}$$
-4 e
$$e^{-4}$$
= 0.0183156388887342
= 0.0183156388887342
The graph