Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (4+x^2)/(-6+2*x)
-
Limit of ((1+x)/(1+2*x))^x
-
Limit of (n/(1+n))^(5+3*n)
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Limit of (9^x-8^x)/asin(3*x)
- Integral of d{x}:
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x^2/2
- Derivative of:
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x^2/2
- Canonical form:
- x^2/2
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Identical expressions
- x^ two / two
- x squared divide by 2
- x to the power of two divide by two
- x2/2
- x²/2
- x to the power of 2/2
- x^2 divide by 2
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Similar expressions
- 3+x^2/2-6*x
- (2+x+x^2)/(2+x^2)
- -35-9*x-39*x^2/2
- (2-x^3+3*x^2)/(2+x^2-x)
- -7+x^2/2+5*x
Limit of the function x^2/2
The solution
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/ 2\
|x |
lim |--|
x->oo\2 /
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right)$$
Limit(x^2/2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{2 \frac{1}{x^{2}}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{2 \frac{1}{x^{2}}} = \lim_{u \to 0^+}\left(\frac{1}{2 u^{2}}\right)$$
=
$$\frac{1}{0 \cdot 2} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{2 \frac{1}{x^{2}}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{2 \frac{1}{x^{2}}} = \lim_{u \to 0^+}\left(\frac{1}{2 u^{2}}\right)$$
=
$$\frac{1}{0 \cdot 2} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right) = \infty$$
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
$$\lim_{x \to \infty}\left(\frac{x^{2}}{2}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x^{2}}{2}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{2}}{2}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x^{2}}{2}\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{2}}{2}\right) = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{2}}{2}\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x^{2}}{2}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{2}}{2}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x^{2}}{2}\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{2}}{2}\right) = \frac{1}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{2}}{2}\right) = \infty$$
More at x→-oo
The graph