Mister Exam
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How to use it?
- Limit of the function:
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Limit of (3+2*n)/|-1+2*n|
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Limit of (3+x^2-4*x)/(-9+x^2)
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Limit of (x^2-3*x)/(-8+x^2)
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Limit of (1+5*x)*(-1+5*x)
- Integral of d{x}:
- 1+y^2
-
Identical expressions
- one +y^ two
- 1 plus y squared
- one plus y to the power of two
- 1+y2
- 1+y²
- 1+y to the power of 2
-
Similar expressions
- 1-y^2
Limit of the function 1+y^2
The solution
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/ 2\ lim \1 + y / y->oo
$$\lim_{y \to \infty}\left(y^{2} + 1\right)$$
Limit(1 + y^2, y, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{y \to \infty}\left(y^{2} + 1\right)$$
Let's divide numerator and denominator by y^2:
$$\lim_{y \to \infty}\left(y^{2} + 1\right)$$ =
$$\lim_{y \to \infty}\left(\frac{1 + \frac{1}{y^{2}}}{\frac{1}{y^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{y}$$
then
$$\lim_{y \to \infty}\left(\frac{1 + \frac{1}{y^{2}}}{\frac{1}{y^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + 1}{u^{2}}\right)$$
=
$$\frac{0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{y \to \infty}\left(y^{2} + 1\right) = \infty$$
$$\lim_{y \to \infty}\left(y^{2} + 1\right)$$
Let's divide numerator and denominator by y^2:
$$\lim_{y \to \infty}\left(y^{2} + 1\right)$$ =
$$\lim_{y \to \infty}\left(\frac{1 + \frac{1}{y^{2}}}{\frac{1}{y^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{y}$$
then
$$\lim_{y \to \infty}\left(\frac{1 + \frac{1}{y^{2}}}{\frac{1}{y^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + 1}{u^{2}}\right)$$
=
$$\frac{0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{y \to \infty}\left(y^{2} + 1\right) = \infty$$
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 y→0, -oo, +oo, 1
$$\lim_{y \to \infty}\left(y^{2} + 1\right) = \infty$$
$$\lim_{y \to 0^-}\left(y^{2} + 1\right) = 1$$
More at y→0 from the left
$$\lim_{y \to 0^+}\left(y^{2} + 1\right) = 1$$
More at y→0 from the right
$$\lim_{y \to 1^-}\left(y^{2} + 1\right) = 2$$
More at y→1 from the left
$$\lim_{y \to 1^+}\left(y^{2} + 1\right) = 2$$
More at y→1 from the right
$$\lim_{y \to -\infty}\left(y^{2} + 1\right) = \infty$$
More at y→-oo
$$\lim_{y \to 0^-}\left(y^{2} + 1\right) = 1$$
More at y→0 from the left
$$\lim_{y \to 0^+}\left(y^{2} + 1\right) = 1$$
More at y→0 from the right
$$\lim_{y \to 1^-}\left(y^{2} + 1\right) = 2$$
More at y→1 from the left
$$\lim_{y \to 1^+}\left(y^{2} + 1\right) = 2$$
More at y→1 from the right
$$\lim_{y \to -\infty}\left(y^{2} + 1\right) = \infty$$
More at y→-oo
The graph