Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-1+cos(x))/x^2
-
Limit of (3*x+14*x^2)/(1+2*x+7*x^2)
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Limit of (-6*tan(x)+2*tan(3*x))/(-atan(3*x)+3*atan(x))
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Limit of (5+x)/(-5+x^2+4*x)
- Integral of d{x}:
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1+x^2
- Derivative of:
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1+x^2
- Graphing y =:
- 1+x^2
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Identical expressions
- one +x^ two
- 1 plus x squared
- one plus x to the power of two
- 1+x2
- 1+x²
- 1+x to the power of 2
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Similar expressions
- 1-x^2
- (1-x^2)/(1+x^2)
- -1+x^2-x^3+2*x
- (-1+x^2)/(-2+x+x^2)
- (-1+x^2)/(-10+2*x)
Limit of the function 1+x^2
The solution
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/ 2\ lim \1 + x / x->oo
$$\lim_{x \to \infty}\left(x^{2} + 1\right)$$
Limit(1 + x^2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{2} + 1\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(x^{2} + 1\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + 1}{u^{2}}\right)$$
=
$$\frac{0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{2} + 1\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{2} + 1\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(x^{2} + 1\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{2}}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{2} + 1}{u^{2}}\right)$$
=
$$\frac{0^{2} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{2} + 1\right) = \infty$$
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 x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(x^{2} + 1\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{2} + 1\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{2} + 1\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{2} + 1\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{2} + 1\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{2} + 1\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{2} + 1\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{2} + 1\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{2} + 1\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{2} + 1\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{2} + 1\right) = \infty$$
More at x→-oo
The graph