Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of 1-cos(3*x)
-
Limit of (-16+x^2-6*x)/(-2+x+x^2)
-
Limit of (-cos(x)^3+cos(x))/(4*x*sin(x))
-
Limit of cos((1+x)/x^3)
- Derivative of:
-
2*x/(1+x)
- Integral of d{x}:
-
2*x/(1+x)
- Graphing y =:
-
2*x/(1+x)
-
Identical expressions
- two *x/(one +x)
- 2 multiply by x divide by (1 plus x)
- two multiply by x divide by (one plus x)
- 2x/(1+x)
- 2x/1+x
- 2*x divide by (1+x)
-
Similar expressions
- 2*x/(1-x)
- (1-e^(5*x)+sin(2*x))/(1+x-cos(x))
- (-3+2*x)/(1+x^2)
- (-1+2*x)/(1+x+x^2)
- 2^x/(1+x^2)
- ((3+2*x)/(1+x))^x
Limit of the function 2*x/(1+x)
The solution
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/ 2*x \ lim |-----| x->oo\1 + x/
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
Limit((2*x)/(1 + x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{2}{1 + \frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{2}{1 + \frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{2}{u + 1}\right)$$
=
$$\frac{2}{1} = 2$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{2}{1 + \frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{2}{1 + \frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{2}{u + 1}\right)$$
=
$$\frac{2}{1} = 2$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right) = 2$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(2 x\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 x}{\frac{d}{d x} \left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty} 2$$
=
$$\lim_{x \to \infty} 2$$
=
$$2$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(2 x\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 x}{\frac{d}{d x} \left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty} 2$$
=
$$\lim_{x \to \infty} 2$$
=
$$2$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{2 x}{x + 1}\right) = 2$$
$$\lim_{x \to 0^-}\left(\frac{2 x}{x + 1}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x}{x + 1}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2 x}{x + 1}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x}{x + 1}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x}{x + 1}\right) = 2$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{2 x}{x + 1}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x}{x + 1}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2 x}{x + 1}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x}{x + 1}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x}{x + 1}\right) = 2$$
More at x→-oo
The graph