Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
- Limit of 2016+965*e^(-168*x)*sin(-115+547*x)
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Limit of (1+n)^(1/3)/n^(1/3)
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Limit of (1+9/x)^(3*x)
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Limit of (19-32*x+9*x^4+16*x^6)/(38-12*x^5+8*x^6+12*x)
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Identical expressions
- (one + nine /x)^(three *x)
- (1 plus 9 divide by x) to the power of (3 multiply by x)
- (one plus nine divide by x) to the power of (three multiply by x)
- (1+9/x)(3*x)
- 1+9/x3*x
- (1+9/x)^(3x)
- (1+9/x)(3x)
- 1+9/x3x
- 1+9/x^3x
- (1+9 divide by x)^(3*x)
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Similar expressions
- (1-9/x)^(3*x)
Limit of the function (1+9/x)^(3*x)
The solution
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[src]
3*x
/ 9\
lim |1 + -|
x->oo\ x/
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x}$$
Limit((1 + 9/x)^(3*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x}$$
transform
do replacement
$$u = \frac{x}{9}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{27 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{27 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{27}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{27} = e^{27}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x} = e^{27}$$
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x}$$
transform
do replacement
$$u = \frac{x}{9}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{27 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{27 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{27}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{27} = e^{27}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x} = e^{27}$$
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
$$\lim_{x \to \infty} \left(1 + \frac{9}{x}\right)^{3 x} = e^{27}$$
$$\lim_{x \to 0^-} \left(1 + \frac{9}{x}\right)^{3 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{9}{x}\right)^{3 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{9}{x}\right)^{3 x} = 1000$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{9}{x}\right)^{3 x} = 1000$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{9}{x}\right)^{3 x} = e^{27}$$
More at x→-oo
$$\lim_{x \to 0^-} \left(1 + \frac{9}{x}\right)^{3 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{9}{x}\right)^{3 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{9}{x}\right)^{3 x} = 1000$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{9}{x}\right)^{3 x} = 1000$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{9}{x}\right)^{3 x} = e^{27}$$
More at x→-oo
The graph