Find Limit of Function: Does 0/0 Exist?

In summary: That's the important thing to remember. You don't have to do that for the denominator. In summary, when evaluating limits, it is important to remember that the indeterminate form 0/0 does not have a definite value and further steps must be taken to determine the limit. One way to do this is to factor both the numerator and denominator and see if there is a common factor that can be cancelled out, as this can often lead to a non-zero limit. Additionally, L'Hopital's Rule can be used in cases with the indeterminate forms 0/0 and ±∞/∞.
  • #1
Ipos Manger
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Homework Statement



Does lim x-> -2 (3x2+ax+a+3)/(x2+x-2) exist?

If so, find the limit.

Homework Equations



-

The Attempt at a Solution



I've tried factorizing the denominator to (x+2)(x-1), but then I don't know how to proceed on the exercise. I have seen that the limit exists when the numerator equals 0, but why is this (the denominator is also 0)? Does 0/0 exist at all?
Code:
 
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  • #2
Ipos Manger said:

Homework Statement



Does lim x-> -2 (3x2+ax+a+3)/(x2+x-2) exist?

If so, find the limit.

Homework Equations



-

The Attempt at a Solution



I've tried factorizing the denominator to (x+2)(x-1), but then I don't know how to proceed on the exercise. I have seen that the limit exists when the numerator equals 0, but why is this (the denominator is also 0)? Does 0/0 exist at all?

What value of a would make the numerator zero when x = -2?

When both numerator and denominator are zero (i.e., the indeterminate form 0/0), it's usually because there is a factor in common between the top and bottom. Although not certain, there's a good chance that a limit exists.
 
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  • #3
Mark44 said:
What value of a would make the numerator zero when x = -2?

When both numerator and denominator are zero (i.e., the indeterminate form 0/0), it's usually because there is a factor in common between the top and bottom. Although not certain, there's a good chance that a limit exists.
a = 15.

So that means I just have to learn that by heart? Isn't there sort of a formal proof to test that? On the other hand, how do you know there's a factor in common between the top and bottom knowing that numerator and denominator are zero?

Thank you for your answer.
 
  • #4
I don't think you need to memorize what I said - just be aware that if a limit has the form "0/0" you're not done yet. "0/0" is not a value - it is an indeterminate form, which means that you can't determine a value. All of the limits below have this form, but they have wildly different limit values.

$$\lim_{x \to 0}\frac {x^2}{x} = 0$$
$$\lim_{x \to 0}\frac {x}{x} = 1$$
$$\lim_{x \to 0^+}\frac {x}{x^2} = \infty $$

Note that in the 3rd limit, the two-sided limit doesn't exist, and that's the reason I am using the one-sided limit, the limit as x approaches zero from the right.

As far as a theorem goes, the only one that comes to mind is L'Hopital's Rule, which is applicable in cases with the indeterminate forms 0/0 and ±∞/∞.
 
  • #5
After you have factored the denominator, factor the numerator (or use Mark44's shortcut).
 
  • #6
Ipos Manger said:
On the other hand, how do you know there's a factor in common between the top and bottom knowing that numerator and denominator are zero?
There's a theorem that says that if p(x) is a non-constant polynomial and p(a)=0, then p(x) can be factored into p(x)=(x-a)q(x), where q(x) is another polynomial. In this problem both the numerator and denominator are polynomials; therefore, if they both equal 0 at x=-2, they have (x-(-2)) as a common factor.
 
  • #7
I'm sorry to hijack the thread, but does this mean that i have to learn as a general rule if the denominator is 0 (in the case of the limit), I have to solve the numerator for 0 and find the values that I can factorize? Thank you.
 
  • #8
chatnay said:
I'm sorry to hijack the thread, but does this mean that i have to learn as a general rule if the denominator is 0 (in the case of the limit), I have to solve the numerator for 0 and find the values that I can factorize? Thank you.

I would say no, you don't have to do that. You must try to factorize the numerator IF it goes to 0.
 

FAQ: Find Limit of Function: Does 0/0 Exist?

What is the definition of a limit of a function?

The limit of a function is the value that a function approaches as its input variable approaches a specific value or point. It is a fundamental concept in calculus and is used to describe the behavior of a function near a certain point.

Can a limit of a function exist if the function is undefined at that point?

Yes, a limit can still exist even if the function is undefined at that point. This is because the limit considers the behavior of the function as the input approaches the point, not the actual value of the function at that point.

How do you determine if a limit of a function exists?

To determine if a limit exists, you can use various methods such as direct substitution, factoring, or algebraic manipulation. If these methods fail, you can use the limit laws or L'Hopital's rule to evaluate the limit.

What does it mean if the limit of a function approaches infinity?

If the limit of a function approaches infinity, it means that the function is increasing without bound as the input approaches a certain point. This can also be written as "the limit does not exist" or "the limit is undefined".

Can a limit of a function exist if the function is discontinuous at that point?

Yes, a limit can still exist even if the function is discontinuous at that point. This is because a function can have a limit at a point even if it is not defined or has a different value at that point.

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