Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (2^(1+2*x)+3^(2+x))/(5+4^(2+x))
-
Limit of 1/(x*(3+x))
-
Limit of 1+x*log(2+x)
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Limit of ((1+x^2)^(1/3)-x*cot(x))/(x*sin(x))
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Identical expressions
- (one + one /(seven *x))^(five *x)
- (1 plus 1 divide by (7 multiply by x)) to the power of (5 multiply by x)
- (one plus one divide by (seven multiply by x)) to the power of (five multiply by x)
- (1+1/(7*x))(5*x)
- 1+1/7*x5*x
- (1+1/(7x))^(5x)
- (1+1/(7x))(5x)
- 1+1/7x5x
- 1+1/7x^5x
- (1+1 divide by (7*x))^(5*x)
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Similar expressions
- (1-1/(7*x))^(5*x)
Limit of the function (1+1/(7*x))^(5*x)
The solution
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5*x
/ 1 \
lim |1 + ---|
x->oo\ 7*x/
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x}$$
Limit((1 + 1/(7*x))^(5*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x}$$
transform
do replacement
then
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{7}}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{7}}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{7}}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{7}} = e^{\frac{5}{7}}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x} = e^{\frac{5}{7}}$$
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x}$$
transform
do replacement
x
u = ---
1/7then
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{7}}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{\frac{5 u}{7}}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{7}}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{\frac{5}{7}} = e^{\frac{5}{7}}$$
The final answer:
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x} = e^{\frac{5}{7}}$$
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
$$\lim_{x \to \infty} \left(1 + \frac{1}{7 x}\right)^{5 x} = e^{\frac{5}{7}}$$
$$\lim_{x \to 0^-} \left(1 + \frac{1}{7 x}\right)^{5 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{1}{7 x}\right)^{5 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{1}{7 x}\right)^{5 x} = \frac{32768}{16807}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{1}{7 x}\right)^{5 x} = \frac{32768}{16807}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{1}{7 x}\right)^{5 x} = e^{\frac{5}{7}}$$
More at x→-oo
$$\lim_{x \to 0^-} \left(1 + \frac{1}{7 x}\right)^{5 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{1}{7 x}\right)^{5 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{1}{7 x}\right)^{5 x} = \frac{32768}{16807}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{1}{7 x}\right)^{5 x} = \frac{32768}{16807}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{1}{7 x}\right)^{5 x} = e^{\frac{5}{7}}$$
More at x→-oo
The graph