Mister Exam
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How to use it?
- Limit of the function:
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Limit of (-5+7*n)/(2+n)
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Limit of (-64+4^x)/(-3+x)
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Limit of (-7+3*x^2+5*x)/(1+x+3*x^2)
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Limit of (1+6*x)^(1/x)
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Identical expressions
- (one + six *x)^(one /x)
- (1 plus 6 multiply by x) to the power of (1 divide by x)
- (one plus six multiply by x) to the power of (one divide by x)
- (1+6*x)(1/x)
- 1+6*x1/x
- (1+6x)^(1/x)
- (1+6x)(1/x)
- 1+6x1/x
- 1+6x^1/x
- (1+6*x)^(1 divide by x)
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Similar expressions
- (1-6*x)^(1/x)
Limit of the function (1+6*x)^(1/x)
The solution
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[src]
x _________ lim \/ 1 + 6*x x->0+
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}}$$
Limit((1 + 6*x)^(1/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{6 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{6}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{6 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{6 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{6}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{6} = e^{6}$$
The final answer:
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}} = e^{6}$$
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}}$$
transform
do replacement
$$u = \frac{1}{6 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{6}{\frac{1}{x}}\right)^{\frac{1}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{6 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{6 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{6}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{6} = e^{6}$$
The final answer:
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}} = e^{6}$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(6 x + 1\right)^{\frac{1}{x}} = e^{6}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}} = e^{6}$$
$$\lim_{x \to \infty} \left(6 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(6 x + 1\right)^{\frac{1}{x}} = 7$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(6 x + 1\right)^{\frac{1}{x}} = 7$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(6 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}} = e^{6}$$
$$\lim_{x \to \infty} \left(6 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(6 x + 1\right)^{\frac{1}{x}} = 7$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(6 x + 1\right)^{\frac{1}{x}} = 7$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(6 x + 1\right)^{\frac{1}{x}} = 1$$
More at x→-oo
One‐sided limits
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x _________ lim \/ 1 + 6*x x->0+
$$\lim_{x \to 0^+} \left(6 x + 1\right)^{\frac{1}{x}}$$
6 e
$$e^{6}$$
= 403.428793492735
x _________ lim \/ 1 + 6*x x->0-
$$\lim_{x \to 0^-} \left(6 x + 1\right)^{\frac{1}{x}}$$
6 e
$$e^{6}$$
exp(6)
The graph