Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (-3+x^2-2*x)/(-3+x)
-
Limit of 5^x-cos(x)
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Limit of (x/2)^(1/(-2+x))
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Limit of (1-3/x)^x
- Derivative of:
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1/(1+x^2)
- Integral of d{x}:
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1/(1+x^2)
- Graphing y =:
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1/(1+x^2)
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Identical expressions
- one /(one +x^ two)
- 1 divide by (1 plus x squared )
- one divide by (one plus x to the power of two)
- 1/(1+x2)
- 1/1+x2
- 1/(1+x²)
- 1/(1+x to the power of 2)
- 1/1+x^2
- 1 divide by (1+x^2)
-
Similar expressions
- 1/(1-x^2)
- 1/(1+x^2+2*x)-1/(-1+x^2)
Limit of the function 1/(1+x^2)
The solution
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1
lim ------
x->oo 2
1 + x
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1}$$
Limit(1/(1 + x^2), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1}$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1}$$ =
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u^{2}}{u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1} = 0$$
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1}$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1}$$ =
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right) = \lim_{u \to 0^+}\left(\frac{u^{2}}{u^{2} + 1}\right)$$
=
$$\frac{0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1} = 0$$
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
$$\lim_{x \to \infty} \frac{1}{x^{2} + 1} = 0$$
$$\lim_{x \to 0^-} \frac{1}{x^{2} + 1} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \frac{1}{x^{2} + 1} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \frac{1}{x^{2} + 1} = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \frac{1}{x^{2} + 1} = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \frac{1}{x^{2} + 1} = 0$$
More at x→-oo
$$\lim_{x \to 0^-} \frac{1}{x^{2} + 1} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \frac{1}{x^{2} + 1} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \frac{1}{x^{2} + 1} = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \frac{1}{x^{2} + 1} = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \frac{1}{x^{2} + 1} = 0$$
More at x→-oo
The graph