Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-1+3*sqrt(x))/(-1+4*sqrt(x))
-
Limit of (-4+2*x)/(6+4*x+7*x^2)
-
Limit of ((5+2*x)/(2*x))^(3*x)
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Limit of ((-4+2*x)/(1+2*x))^(5+3*x)
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Identical expressions
- four /(one +x^ two)
- 4 divide by (1 plus x squared )
- four divide by (one plus x to the power of two)
- 4/(1+x2)
- 4/1+x2
- 4/(1+x²)
- 4/(1+x to the power of 2)
- 4/1+x^2
- 4 divide by (1+x^2)
-
Similar expressions
- 4/(1-x^2)
Limit of the function 4/(1+x^2)
The solution
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/ 4 \
lim |------|
x->oo| 2|
\1 + x /
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right)$$
Limit(4/(1 + x^2), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{4 \frac{1}{x^{2}}}{1 + \frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{4 \frac{1}{x^{2}}}{1 + \frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{4 u^{2}}{u^{2} + 1}\right)$$
=
$$\frac{4 \cdot 0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{4 \frac{1}{x^{2}}}{1 + \frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{4 \frac{1}{x^{2}}}{1 + \frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{4 u^{2}}{u^{2} + 1}\right)$$
=
$$\frac{4 \cdot 0^{2}}{0^{2} + 1} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right) = 0$$
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
$$\lim_{x \to \infty}\left(\frac{4}{x^{2} + 1}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{4}{x^{2} + 1}\right) = 4$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4}{x^{2} + 1}\right) = 4$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4}{x^{2} + 1}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4}{x^{2} + 1}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4}{x^{2} + 1}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{4}{x^{2} + 1}\right) = 4$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4}{x^{2} + 1}\right) = 4$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4}{x^{2} + 1}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4}{x^{2} + 1}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4}{x^{2} + 1}\right) = 0$$
More at x→-oo
The graph