# Limit of (x^3-2x^2-9)/(x^2-2x-3) as x->3 | Homework Help

• asd1249jf
In summary, the limit of (x^3-2x^2-9)/(x^2-2x-3) as x approaches 3 can be found by factoring the numerator and denominator and simplifying, where the numerator has a factor of (x-3) and the denominator has a factor of (x-3). This results in the limit being equal to the limit of (x+3) as x approaches 3, which is equal to 6.
asd1249jf

## Homework Statement

Find the limit of

$\frac{x^3-2x^2-9}{x^2-2x-3}$

as x->3

## The Attempt at a Solution

You factor the bottom portion and top portion, then it looks something like this

$\frac{x(x^2-2x)-9}{(x-3)(x+1)}$

I feel like I can go further about eliminating the demonimator but I don't know what

Is 3 a root of the polynomial x^3- 2x^2 - 9 ? If so, what is then factoring of this polynomial ?

(x - 3) is a factor of the numerator. You can use either synthetic division or plain old polynomial division to find the other factor.

... or notice that

$x^3-2x^2-9 = x^3-3x^2+x^2-9$

and factor by grouping.

l46kok said:

## Homework Statement

Find the limit of

$\frac{x^3-2x^2-9}{x^2-2x-3}$

as x->3

## The Attempt at a Solution

You factor the bottom portion and top portion, then it looks something like this

$\frac{x(x^2-2x)-9}{(x-3)(x+1)}$
No, you did NOT factor the numerator. That is not what "factor" means.

I assume you tried first just putting x= 3 into the fraction and found that both numerator and denominator were 0 when x= 3. The fact that the numerator was 0 tells you that it has a factor of x- 3. $x^3- 2x^2- 9= (x- 3)(ax^2+ bx+ c)$
It shouldn't be hard to see what a, b, and c must be.

I feel like I can go further about eliminating the demonimator but I don't know what

## 1. What is a limit in calculus?

A limit in calculus is a fundamental concept that describes the behavior of a function as its input approaches a specific value. It is denoted by the notation "lim" and is used to determine the value that a function approaches as its input gets closer and closer to a certain value.

## 2. How do you find the limit of a function?

To find the limit of a function, you can use several methods such as direct substitution, factoring, and algebraic manipulation. You can also use the limit laws, which state that the limit of a sum, difference, product, or quotient of two functions is equal to the sum, difference, product, or quotient of their individual limits.

## 3. What does "x->3" mean in the given expression?

The notation "x->3" indicates that the value of x is approaching the number 3. In this context, it means that we are determining the limit of the function as x gets closer and closer to 3.

## 4. Why is it necessary to find the limit of a function?

Finding the limit of a function is necessary because it helps us understand the behavior of the function at a specific point. It also allows us to determine if the function is continuous at that point, which is essential in many applications of calculus.

## 5. Can the limit of a function be undefined?

Yes, the limit of a function can be undefined. This can happen if the function has a vertical asymptote or if the left- and right-hand limits approach different values at a particular point. In the given expression, if the denominator becomes 0 as x approaches 3, the limit will be undefined.

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