Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (e^x-e)/(-1+x)
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Limit of (1-3*x)^(1/x)
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Limit of pi/(x*cot(pi*x/2))
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Limit of sin(7*x)/x^2
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Identical expressions
- (one - three *x)^(one /x)
- (1 minus 3 multiply by x) to the power of (1 divide by x)
- (one minus three multiply by x) to the power of (one divide by x)
- (1-3*x)(1/x)
- 1-3*x1/x
- (1-3x)^(1/x)
- (1-3x)(1/x)
- 1-3x1/x
- 1-3x^1/x
- (1-3*x)^(1 divide by x)
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Similar expressions
- (1+3*x)^(1/x)
Limit of the function (1-3*x)^(1/x)
The solution
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[src]
x _________ lim \/ 1 - 3*x x->0+
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}}$$
Limit((1 - 3*x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{\left(-3\right) x}$$
then
$$\lim_{x \to 0^+} \left(1 - \frac{3}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 3 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 3 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-3}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-3} = e^{-3}$$
The final answer:
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}$$
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{\left(-3\right) x}$$
then
$$\lim_{x \to 0^+} \left(1 - \frac{3}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 3 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 3 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-3}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-3} = e^{-3}$$
The final answer:
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}$$
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 - 3*x x->0+
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}}$$
-3 e
$$e^{-3}$$
= 0.0497870683678639
x _________ lim \/ 1 - 3*x x->0-
$$\lim_{x \to 0^-} \left(1 - 3 x\right)^{\frac{1}{x}}$$
-3 e
$$e^{-3}$$
= 0.0497870683678639
= 0.0497870683678639
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}$$
$$\lim_{x \to \infty} \left(1 - 3 x\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(1 - 3 x\right)^{\frac{1}{x}} = -2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 - 3 x\right)^{\frac{1}{x}} = -2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 - 3 x\right)^{\frac{1}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}$$
$$\lim_{x \to \infty} \left(1 - 3 x\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(1 - 3 x\right)^{\frac{1}{x}} = -2$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 - 3 x\right)^{\frac{1}{x}} = -2$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 - 3 x\right)^{\frac{1}{x}} = 1$$
More at x→-oo
The graph