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