Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of cos(x^2)
-
Limit of (-2+cos(3*x)+cosh(3*x))/(x*(-sin(x)+tan(x)))
-
Limit of (-cos(8*x)+cos(2*x))/(x*tan(x))
-
Limit of x^2*atan(3*x)^2/4
- Derivative of:
- (1+1/x)^3
-
Identical expressions
- (one + one /x)^ three
- (1 plus 1 divide by x) cubed
- (one plus one divide by x) to the power of three
- (1+1/x)3
- 1+1/x3
- (1+1/x)³
- (1+1/x) to the power of 3
- 1+1/x^3
- (1+1 divide by x)^3
-
Similar expressions
- (1-1/x)^3
Limit of the function (1+1/x)^3
The solution
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3
/ 1\
lim |1 + -|
x->oo\ x/
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{3}$$
Limit((1 + 1/x)^3, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x^{3} + 3 x^{2} + 3 x + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x^{3} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{3}$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{3}}{x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{3} + 3 x^{2} + 3 x + 1\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 x^{2} + 6 x + 3}{3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x^{2} + 6 x + 3\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6 x + 6}{6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(6 x + 6\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to \infty} 1$$
=
$$\lim_{x \to \infty} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x^{3} + 3 x^{2} + 3 x + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x^{3} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{3}$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{3}}{x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{3} + 3 x^{2} + 3 x + 1\right)}{\frac{d}{d x} x^{3}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3 x^{2} + 6 x + 3}{3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x^{2} + 6 x + 3\right)}{\frac{d}{d x} 3 x^{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6 x + 6}{6 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(6 x + 6\right)}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to \infty} 1$$
=
$$\lim_{x \to \infty} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{3} = 1$$
$$\lim_{x \to 0^-} \left(1 + \frac{1}{x}\right)^{3} = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{1}{x}\right)^{3} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{1}{x}\right)^{3} = 8$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{1}{x}\right)^{3} = 8$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{1}{x}\right)^{3} = 1$$
More at x→-oo
$$\lim_{x \to 0^-} \left(1 + \frac{1}{x}\right)^{3} = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(1 + \frac{1}{x}\right)^{3} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(1 + \frac{1}{x}\right)^{3} = 8$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(1 + \frac{1}{x}\right)^{3} = 8$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(1 + \frac{1}{x}\right)^{3} = 1$$
More at x→-oo
The graph