Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-2*n^2+4*n+7*n^3)/(5+2*n^3)
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Limit of (1-cos(2*x))/(1-cos(4*x))
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Limit of (1-4/x)^(7*x)
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Limit of (-1+(3+2*n)^2)/(-1+(1+2*n)^2)
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Identical expressions
- (one - four /x)^(seven *x)
- (1 minus 4 divide by x) to the power of (7 multiply by x)
- (one minus four divide by x) to the power of (seven multiply by x)
- (1-4/x)(7*x)
- 1-4/x7*x
- (1-4/x)^(7x)
- (1-4/x)(7x)
- 1-4/x7x
- 1-4/x^7x
- (1-4 divide by x)^(7*x)
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Similar expressions
- (1+4/x)^(7*x)
Limit of the function (1-4/x)^(7*x)
The solution
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7*x
/ 4\
lim |1 - -|
x->oo\ x/
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x}$$
Limit((1 - 4/x)^(7*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x}$$
transform
do replacement
$$u = \frac{x}{-4}$$
then
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 28 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 28 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-28}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-28} = e^{-28}$$
The final answer:
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x} = e^{-28}$$
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x}$$
transform
do replacement
$$u = \frac{x}{-4}$$
then
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x}$$ =
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 28 u}$$
=
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{- 28 u}$$
=
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-28}$$
The limit
$$\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to \infty} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-28} = e^{-28}$$
The final answer:
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x} = e^{-28}$$
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
$$\lim_{x \to \infty} \left(1 - \frac{4}{x}\right)^{7 x} = e^{-28}$$
$$\lim_{x \to 0^-} \left(1 - \frac{4}{x}\right)^{7 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 - \frac{4}{x}\right)^{7 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 - \frac{4}{x}\right)^{7 x} = -2187$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 - \frac{4}{x}\right)^{7 x} = -2187$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 - \frac{4}{x}\right)^{7 x} = e^{-28}$$
More at x→-oo
$$\lim_{x \to 0^-} \left(1 - \frac{4}{x}\right)^{7 x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 - \frac{4}{x}\right)^{7 x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 - \frac{4}{x}\right)^{7 x} = -2187$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 - \frac{4}{x}\right)^{7 x} = -2187$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 - \frac{4}{x}\right)^{7 x} = e^{-28}$$
More at x→-oo
The graph