Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of -pi+(1+cos(x))/x
-
Limit of (1-5*x)^(1/x)
-
Limit of (1+2/n)^(4*n)
- Derivative of:
-
x+x^4
- Factor polynomial:
- x+x^4
-
Identical expressions
- x+x^ four
- x plus x to the power of 4
- x plus x to the power of four
- x+x4
- x+x⁴
-
Similar expressions
- x-x^4
- x+x^4+6*x^2
- 2/(2+x+x^4)
Limit of the function x+x^4
The solution
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/ 4\ lim \x + x / x->oo
$$\lim_{x \to \infty}\left(x^{4} + x\right)$$
Limit(x + x^4, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x^{4} + x\right)$$
Let's divide numerator and denominator by x^4:
$$\lim_{x \to \infty}\left(x^{4} + x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{3}}}{\frac{1}{x^{4}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{3}}}{\frac{1}{x^{4}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{3} + 1}{u^{4}}\right)$$
=
$$\frac{0^{3} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{4} + x\right) = \infty$$
$$\lim_{x \to \infty}\left(x^{4} + x\right)$$
Let's divide numerator and denominator by x^4:
$$\lim_{x \to \infty}\left(x^{4} + x\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{3}}}{\frac{1}{x^{4}}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 + \frac{1}{x^{3}}}{\frac{1}{x^{4}}}\right) = \lim_{u \to 0^+}\left(\frac{u^{3} + 1}{u^{4}}\right)$$
=
$$\frac{0^{3} + 1}{0} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(x^{4} + x\right) = \infty$$
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
$$\lim_{x \to \infty}\left(x^{4} + x\right) = \infty$$
$$\lim_{x \to 0^-}\left(x^{4} + x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{4} + x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{4} + x\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{4} + x\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{4} + x\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x^{4} + x\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x^{4} + x\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x^{4} + x\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x^{4} + x\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x^{4} + x\right) = \infty$$
More at x→-oo
The graph