# Finding the Limit of x-> ∞ (5x^2-1)/(x^2)

• grace77
In summary, the problem is finding the limit of the rational expression (5x^2-1)/(x^2) as x approaches infinity. The correct way to solve this is by splitting up the expression, cancelling out any common factors, and then evaluating the limits of the resulting expressions. In this case, the limit would be equal to 5.
grace77
Problem statement
Find lim x-> infinity

(5x^2-1)/(x^2)

Revelant equations
None
Attempt at a solution
Just wanted to make sure I did this right.

Since the degree of the numerator is equal to the denominator does that mean that the limit is just the numerical coefficient of the leading term in the numerator and denominator so in this case it would be 5?

Seat of the pants says so, but the proper way to show the limit is to split up the rational expression, do the necessary cancellations, and then evaluate the limits of the resulting expressions.

SteamKing said:
Seat of the pants says so, but the proper way to show the limit is to split up the rational expression, do the necessary cancellations, and then evaluate the limits of the resulting expressions.
In this case if you spilt it up you would get 5(infinity)-1/infinity ?

grace77 said:
In this case if you spilt it up you would get 5(infinity)-1/infinity ?

No, never. You want the limit of ##f(x) = (5 x^2 - 1)/x^2##. This can be written as
$$f(x) = \frac{5 x^2}{x^2} - \frac{1}{x^2}$$
Do you see now what happens?

Ray Vickson said:
No, never. You want the limit of ##f(x) = (5 x^2 - 1)/x^2##. This can be written as

$$f(x) = \frac{5 x^2}{x^2} - \frac{1}{x^2}$$

Do you see now what happens?
Yes I see it now! It would be equal to 5-0=5!

## 1. What is the limit of (5x^2-1)/(x^2) as x approaches infinity?

The limit of (5x^2-1)/(x^2) as x approaches infinity is equal to 5. This can be determined by dividing the highest power of x in the numerator and denominator, which in this case is x^2. As x approaches infinity, the other terms become negligible and the limit simplifies to 5.

## 2. How do you find the limit of a function as x approaches infinity?

To find the limit of a function as x approaches infinity, you need to analyze the highest powers of x in the numerator and denominator. If the highest power in the numerator is greater than the highest power in the denominator, the limit will be infinity. If the highest powers are equal, the limit can be calculated by dividing the coefficients of those powers. If the highest power in the denominator is greater, the limit will be zero.

## 3. Can the limit of a function as x approaches infinity be a negative number?

Yes, the limit of a function as x approaches infinity can be a negative number. This can happen when there is a negative sign in front of the highest power of x in the numerator, making the overall limit negative.

## 4. What is the difference between a finite limit and an infinite limit?

A finite limit is a numerical value that a function approaches as the independent variable (x) approaches a specific value. An infinite limit occurs when the function approaches either positive or negative infinity as x approaches a specific value.

## 5. In what situations does a function not have a limit as x approaches infinity?

A function does not have a limit as x approaches infinity if the function oscillates between two values or if the limit goes to infinity in different directions from each side of the x-axis. It can also not have a limit if the function has a vertical asymptote at infinity.

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