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