Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of sin(2*x)/sin(x)
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Limit of (1+4/x)^(3*x)
-
Limit of (-2+sqrt(4+x))/(3*atan(x))
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Limit of (2-9*x+9*x^2)/(9+4*x)
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Identical expressions
- (one + four /x)^(three *x)
- (1 plus 4 divide by x) to the power of (3 multiply by x)
- (one plus four divide by x) to the power of (three multiply by x)
- (1+4/x)(3*x)
- 1+4/x3*x
- (1+4/x)^(3x)
- (1+4/x)(3x)
- 1+4/x3x
- 1+4/x^3x
- (1+4 divide by x)^(3*x)
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Similar expressions
- (1-4/x)^(3*x)
Limit of the function (1+4/x)^(3*x)
The solution
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[src]
3*x
/ 4\
lim |1 + -|
x->oo\ x/
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}$$
Limit((1 + 4/x)^(3*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}$$
transform
do replacement
$$u = \frac{x}{4}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{12 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{12 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{12}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{12} = e^{12}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}$$
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}$$
transform
do replacement
$$u = \frac{x}{4}$$
then
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{12 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{12 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{12}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{12} = e^{12}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}$$
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
$$\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}$$
$$\lim_{x \to 0^-} \left(1 + \frac{4}{x}\right)^{3 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{4}{x}\right)^{3 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{4}{x}\right)^{3 x} = 125$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{4}{x}\right)^{3 x} = 125$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}$$
More at x→-oo
$$\lim_{x \to 0^-} \left(1 + \frac{4}{x}\right)^{3 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{4}{x}\right)^{3 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{4}{x}\right)^{3 x} = 125$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{4}{x}\right)^{3 x} = 125$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{4}{x}\right)^{3 x} = e^{12}$$
More at x→-oo
The graph