Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-log(1+x^2)+sin(x))/(2-e^(3*x)-sqrt(1+x))
-
Limit of sin(2*x)/log(1+x)
-
Limit of sin(21*x)/(3*x)
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Limit of (-2*sin(x)+sin(2*x))/x
- Factor polynomial:
- x+3*x^2
-
Identical expressions
- x+ three *x^ two
- x plus 3 multiply by x squared
- x plus three multiply by x to the power of two
- x+3*x2
- x+3*x²
- x+3*x to the power of 2
- x+3x^2
- x+3x2
-
Similar expressions
- (-4+x^2)/(-10+x+3*x^2)
- (-16+x^2)/(16-16*x+3*x^2)
- -15-4*x+3*x^2
- 1-2*x+3*x^2
- x-3*x^2
Limit of the function x+3*x^2
The solution
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/ 2\ lim \x + 3*x / x->oo
$$\lim_{x \to \infty}\left(3 x^{2} + x\right)$$
Limit(x + 3*x^2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(3 x^{2} + x\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(3 x^{2} + x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{3 + \frac{1}{x}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{3 + \frac{1}{x}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u + 3}{u^{2}}\right)$$
=
$$\frac{3}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(3 x^{2} + x\right) = \infty$$
$$\lim_{x \to \infty}\left(3 x^{2} + x\right)$$
Let's divide numerator and denominator by x^2:
$$\lim_{x \to \infty}\left(3 x^{2} + x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{3 + \frac{1}{x}}{\frac{1}{x^{2}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{3 + \frac{1}{x}}{\frac{1}{x^{2}}}\right) = \lim_{u \to 0^+}\left(\frac{u + 3}{u^{2}}\right)$$
=
$$\frac{3}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(3 x^{2} + x\right) = \infty$$
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
$$\lim_{x \to \infty}\left(3 x^{2} + x\right) = \infty$$
$$\lim_{x \to 0^-}\left(3 x^{2} + x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(3 x^{2} + x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(3 x^{2} + x\right) = 4$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(3 x^{2} + x\right) = 4$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(3 x^{2} + x\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(3 x^{2} + x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(3 x^{2} + x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(3 x^{2} + x\right) = 4$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(3 x^{2} + x\right) = 4$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(3 x^{2} + x\right) = \infty$$
More at x→-oo
The graph