Limit Question: Finding the Limit of (x^2-16)/(4-x) as x Approaches 4

[Johnyi]

Homework Statement

Lim x^2-16/4-x
x→4

Homework Equations

The Attempt at a Solution

I factored it, and ended up with (x-4)(x+4)/(4-x)

Cant cancel anything out, so I plug in 4 back into it, and end up with 8 as the limit. But the book says that the answer is -8?

 

[SammyS]

Homework Statement

Lim x^2-16/4-x
x→4

Homework Equations

The Attempt at a Solution

I factored it, and ended up with (x-4)(x+4)/(4-x)

Cant cancel anything out, so I plug in 4 back into it, and end up with 8 as the limit. But the book says that the answer is -8?

What is -(x-4) ?

 

[Johnyi]

What is -(x-4) ?

Wouldnt that just be 0? Also why is there a - before the (x-4)?

 

[Mark44]

What is -(x-4) ?

Wouldnt that just be 0? Also why is there a - before the (x-4)?

Why do you think it would be zero?

Hint: Distributive property.

As a side note, if you have a hard time understanding how -(x - 4) might possibly be related to 4 - x, I think you will have a really hard time in more advanced topics such as limits.

 

[SammyS]

What is -(x-4) ?

I'm not asking you to take the limit of -(x-4).

Simplify -(x-4).

Alternatively, factor a (-1) out of (x-4).

 

[Johnyi]

Where did you get the - before the (x-4) though?

 

[SammyS]

Where did you get the - before the (x-4) though?

Don't worry about that until you answer the question I asked.

It may be clear to you after that.

 

[Johnyi]

Don't worry about that until you answer the question I asked.

It may be clear to you after that.

So if i distribute the - it will be -x+4, which cancels with the denominator, and I am left with x+4. I plug in 4 for x and get 8?

 

[SammyS]

So if i distribute the - it will be -x+4, which cancels with the denominator, and I am left with x+4. I plug in 4 for x and get 8?

Almost correct.

Yes, you're right about -(x-4) = -x + 4. Of course that is the same as 4 - x .

So rewrite your expression with a denominator of -(x - 4).

Do you see why I had the "-" sign out front of the (x-4) in my previous posts?

 

[Johnyi]

Almost correct.

Yes, you're right about -(x-4) = -x + 4. Of course that is the same as 4 - x .

So rewrite your expression with a denominator of -(x - 4).

Do you see why I had the "-" sign out front of the (x-4) in my previous posts?

Sorry..I just can't understand how the "-" sign came out of nowhere, and why i have to rewrite the denominator as -(x-4)

 

[SammyS]

Sorry..I just can't understand how the "-" sign came out of nowhere, and why i have to rewrite the denominator as -(x-4)

Didn't you determine that -(x-4) is -x + 4 ?

Well, that's the same as 4 - x, which is the denominator.

4 - x is the same as -(x - 4).

So, replace the denominator, which is 4 - x, with -(x - 4) because they're equivalent.

In general, a - b = -(b - a) .

 

[Johnyi]

So then wouldn't they both cancel anyways leaving me with just (x+4)?

 

[SammyS]

So then wouldn't they both cancel anyways leaving me with just (x+4)?

If the denominator is -(x-4) then there is also a "-" sign left uncancelled.