Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-cos(x)^3+cos(x))/(3*x^2)
-
Limit of -1-cos(x)+cosh(x)
-
Limit of (9-x^3)/(1+2*x^3)
-
Limit of (-7+8*x^5)/(-3+2*x^5)
- Integral of d{x}:
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1/(3*x)
- Derivative of:
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1/(3*x)
-
Identical expressions
- one /(three *x)
- 1 divide by (3 multiply by x)
- one divide by (three multiply by x)
- 1/(3x)
- 1/3x
- 1 divide by (3*x)
-
Similar expressions
- cos(x^(1/3))^(x^(-2/3))
- (1-2*x^2)^(1/(3*x^2))
- (-1/2+2^(1/3)*x^(1/3)/2)/(sqrt(1/2+x)-sqrt(2)*sqrt(x))
- -1/(3*x^2)
- 1/(3*x^2)
Limit of the function 1/(3*x)
The solution
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1 lim --- x->oo3*x
$$\lim_{x \to \infty} \frac{1}{3 x}$$
Limit(1/(3*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty} \frac{1}{3 x}$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty} \frac{1}{3 x}$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} \frac{1}{x}}{1}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} \frac{1}{x}}{1}\right) = \lim_{u \to 0^+}\left(\frac{u}{3}\right)$$
=
$$\frac{0}{3} = 0$$
The final answer:
$$\lim_{x \to \infty} \frac{1}{3 x} = 0$$
$$\lim_{x \to \infty} \frac{1}{3 x}$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty} \frac{1}{3 x}$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} \frac{1}{x}}{1}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{1}{3} \frac{1}{x}}{1}\right) = \lim_{u \to 0^+}\left(\frac{u}{3}\right)$$
=
$$\frac{0}{3} = 0$$
The final answer:
$$\lim_{x \to \infty} \frac{1}{3 x} = 0$$
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
$$\lim_{x \to \infty} \frac{1}{3 x} = 0$$
$$\lim_{x \to 0^-} \frac{1}{3 x} = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+} \frac{1}{3 x} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \frac{1}{3 x} = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \frac{1}{3 x} = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \frac{1}{3 x} = 0$$
More at x→-oo
$$\lim_{x \to 0^-} \frac{1}{3 x} = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+} \frac{1}{3 x} = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-} \frac{1}{3 x} = \frac{1}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \frac{1}{3 x} = \frac{1}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \frac{1}{3 x} = 0$$
More at x→-oo
The graph