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- y+x^2
-
Identical expressions
- y+x^ two
- y plus x squared
- y plus x to the power of two
- y+x2
- y+x²
- y+x to the power of 2
-
Similar expressions
- y-x^2
Limit of the function y+x^2
The solution
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/ 2\ lim \y + x / x->oo
$$\lim_{x \to \infty}\left(x^{2} + y\right)$$
Limit(y + x^2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{2} + y\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(x^{2} + y\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{y}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{y}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} y + 1}{u^{2}}\right)$$
=
$$\frac{0^{2} y + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{2} + y\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{2} + y\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(x^{2} + y\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{y}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{y}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} y + 1}{u^{2}}\right)$$
=
$$\frac{0^{2} y + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{2} + y\right) = \infty$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x^{2} + y\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{2} + y\right) = y$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{2} + y\right) = y$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{2} + y\right) = y + 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{2} + y\right) = y + 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{2} + y\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{2} + y\right) = y$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{2} + y\right) = y$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{2} + y\right) = y + 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{2} + y\right) = y + 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{2} + y\right) = \infty$$
More at x→-oo