# Calculating limits to infinity

• Cacophony
In summary, the conversation discusses the calculation of two different limits using L'Hospital's rule. The first limit is incorrectly written as (lnx)^5/x, when it should be (lnx)^{5/x}. The second limit is correctly calculated as 0 using the rule.
Cacophony

## Homework Statement

Can someone tell if if these look right?

none

## The Attempt at a Solution

1. lim (lnx)^5/x =
x->infinity

5lnx/x = (5lnx/x)/(x/x)=

(5lnx/x)/1 = 0/1 = 0

2. lim sinx/x^2=
x->infinity

(sinx/x^2)/(x^2/x^2)=

(sinx/x*x)/1=

(sinx/x)*(1/x)/1 = (sinx/x * 0)/1 = 0

(lnx)^5 isn't equal 5*ln(x). That doesn't look right. What technique are you trying to use here? Or is there just a notational problem?

I thought you could do that with natural logs?

Cacophony said:
I thought you could do that with natural logs?

ln(x^5)=5ln(x). ln(x)^5 isn't equal to 5ln(x). They are two different things. If you meant the first thing you should use parentheses differently.

Last edited:
Oh, so how would i calculate it then?

Cacophony said:
1. lim (lnx)^5/x =
x->infinity

Cacophony,

apparently you meant:

$$\displaystyle\lim_{x\to \infty} \ \bigg[ln(x)\bigg]^{\dfrac{5}{x}}$$

No, i meant ((lnx)^5)/(x)

Cacophony said:
2. lim sinx/x^2=
x->infinity

(sinx/x^2)/(x^2/x^2)=

(sinx/x*x)/1=

(sinx/x)*(1/x)/1 = (sinx/x * 0)/1 = 0

Yes, that's correct, but your line of (sinx/x^2)/(x^2/x^2) was unnecessary. You should immediately split up $$\frac{\sin(x)}{x^2}=\frac{\sin(x)}{x} \cdot \frac{1}{x}$$ as so.
Cacophony said:
No, i meant ((lnx)^5)/(x)
L'Hospital's rule?

## 1. What is a limit to infinity?

A limit to infinity is a mathematical concept that describes the behavior of a function as its input approaches infinity. It represents the value that the function approaches as the input gets larger and larger.

## 2. How do you calculate a limit to infinity?

To calculate a limit to infinity, you can use various techniques such as algebraic manipulation, substitution, and L'Hôpital's rule. The specific method used will depend on the type of function and the given limit.

## 3. What is L'Hôpital's rule?

L'Hôpital's rule is a mathematical theorem used to evaluate limits of indeterminate forms, where both the numerator and denominator of a fraction approach zero or infinity. It states that the limit of the fraction is equal to the limit of the derivatives of the numerator and denominator.

## 4. Can a limit to infinity have a finite value?

Yes, a limit to infinity can have a finite value if the function approaches a specific value as the input approaches infinity. This is known as a horizontal asymptote.

## 5. What is the significance of calculating limits to infinity?

Calculating limits to infinity is important in many areas of mathematics and science, as it allows us to understand and analyze the behavior of functions at extremely large values. It is also used in calculus to find derivatives and integrals of functions.

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