Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (5-3*x^2-2*x)/(3+x+x^2)
-
Limit of (1+4*x)^(1/x)
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Limit of (1+3/x)^(3*x)
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Limit of (4+x^2)/(-6+2*x)
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Identical expressions
- (one + three /x)^(three *x)
- (1 plus 3 divide by x) to the power of (3 multiply by x)
- (one plus three divide by x) to the power of (three multiply by x)
- (1+3/x)(3*x)
- 1+3/x3*x
- (1+3/x)^(3x)
- (1+3/x)(3x)
- 1+3/x3x
- 1+3/x^3x
- (1+3 divide by x)^(3*x)
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Similar expressions
- (1-3/x)^(3*x)
Limit of the function (1+3/x)^(3*x)
The solution
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3*x
/ 3\
lim |1 + -|
x->oo\ x/
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}$$
Limit((1 + 3/x)^(3*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}$$
transform
do replacement
$$u = \frac{x}{3}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{9 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{9 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{9}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{9} = e^{9}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x} = e^{9}$$
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}$$
transform
do replacement
$$u = \frac{x}{3}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{9 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{9 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{9}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{9} = e^{9}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{3 x} = e^{9}$$
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
The graph