Evaluating Limits Involving Constants

[physstudent1]

Homework Statement

Evaluate the limits in the terms of the constants involved:


\lim_{x \rightarrow a} \frac{(x+a)^2-4x^2}{x-a}

Homework Equations

The Attempt at a Solution

First I plugged in a to make sure it was indeterminate and it was it ends up being 0/0.

So I tried to break this up into the difference of two limits;


\lim_{x \rightarrow a} \frac{(x+a)^2}{x-a} -\lim_{x \rightarrow a} \frac{4x^2}{x-a}

but I again got stuck and I'm not even sure if this attempt is the correct way to go, I also tried to split the numerator into the difference of two squares but that lead to nothing, I'm not really sure where to go from here I'd appreciate it if someone could point me in the right direction.

 

[Hurkyl]

I'm not even sure if this attempt is the correct way to go,

Well, as the limit theorems say... if at least one of those two limits exist, then this is a valid way to compute the oriignal limit. Does either one exist?

 

[physstudent1]

I would think they both do although they get 0 in the denominator they do not get 0 in the numerator nor infinity

 

[Hurkyl]

I would think they both do although they get 0 in the denominator they do not get 0 in the numerator nor infinity

Well, what do the limit theorems say about the limit of a fraction when the limit of the numerator is nonzero and the limit of the denominator is zero?

 

[Hurkyl]

Hrm. I suppose it depends on how you are interpreting "exist". Certainly, the limit theorems tell you that the limit is not finite -- so if they exist, the only remaining possibilities are positive or negative infinity. So, there are very few possibilities; can you tell if these limits are +infinity, -infinity, or nonexistant?


(My previous statements were made according to the interpretation that "doesn't exist" includes the csaes where the limit goes to +infinity or to -infinity)

 

[Hurkyl]

Hrm. The limit theorems I remember include cases like:

If \lim_{x \rightarrow a} f(x) exists and is nonzero, and \lim_{x \rightarrow a} g(x) = 0, then \lim_{x \rightarrow a} f(x) / g(x) is either infinite or nonexistant.​

 

[physstudent1]

i know that if you had actual numbers instead of a I could tell if if they were going to infinity but with a variable its confusing me

 

[physstudent1]

does it approach -infinity from the left but +infinity from the right won't the top stay 0 near the limit because near the limit x will be very close to the value of a and you will get 4a^2-4a^2

 

[physstudent1]

waitttt hold on can't you use long division for these types of problems I am looking through my old calculus notebook and I see that I used long division for some

 

[Hurkyl]

does it approach -infinity from the left but +infinity from the right won't the top stay 0 near the limit because near the limit x will be very close to the value of a and you will get 4a^2-4a^2

waitttt hold on can't you use long division for these types of problems I am looking through my old calculus notebook and I see that I used long division for some

You're right on both counts! You have both determined why your approach won't work, and you have recalled an approach that will work!

 

[physstudent1]

:_) is the answer -4a

 

[Hurkyl]

I agree with that.

 

[physstudent1]

thanks for all the help hurkyl