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