Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (1-4*x)^(1/x)
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Limit of (-16+x^2+6*x)/(-2-5*x+3*x^2)
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Limit of (1+x)^(2/3)-(-1+x)^(2/3)
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Limit of 1/3+x/3
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Identical expressions
- one / three +x/ three
- 1 divide by 3 plus x divide by 3
- one divide by three plus x divide by three
- 1 divide by 3+x divide by 3
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Similar expressions
- 1/3-x/3
Limit of the function 1/3+x/3
The solution
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/1 x\ lim |- + -| x->oo\3 3/
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right)$$
Limit(1/3 + x/3, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} + \frac{1}{3 x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} + \frac{1}{3 x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{\frac{u}{3} + \frac{1}{3}}{u}\right)$$
=
$$\frac{\frac{0}{3} + \frac{1}{3}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} + \frac{1}{3 x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} + \frac{1}{3 x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{\frac{u}{3} + \frac{1}{3}}{u}\right)$$
=
$$\frac{\frac{0}{3} + \frac{1}{3}}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right) = \infty$$
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
$$\lim_{x \to \infty}\left(\frac{x}{3} + \frac{1}{3}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{1}{3}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{2}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{2}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{3} + \frac{1}{3}\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{1}{3}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{2}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{3} + \frac{1}{3}\right) = \frac{2}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{3} + \frac{1}{3}\right) = -\infty$$
More at x→-oo
The graph