Determine the value of a for a given function.

[Torshi]

Determine the value of "a" for a given function.

Homework Statement

Consider the following function: f(x) { ax+7, x<2 and 4x-a, x≥2

A.) Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8. You mus demonstrate all limits.

Homework Equations

ax+7, x<2 and 4x-a, x≥2

The Attempt at a Solution

Approaching from left I got lim x->2- (2a+7) and from right lim x->2+ (8-a)
Am I trying to find a? If I set both limits equal I get a = .33, but the question asks determine the value of such that lim x->2- f(x) = 8 ? That would mean a =1/2 because 2* 1/2 = 1 + 7 = 8, but I don't think that's right unless I'm reading the question wrong..


Edit:
Or do I just state that ax+7 since for this question says lim x->2- (approaching) left f(x) = 8 do I say coming from the left is 8 while from the right it's still 8-a then set those two equal to calculate for "a"? Getting -1 as my answer for "a"?

 

[Mark44]

Homework Statement

Consider the following function: f(x) { ax+7, x<2 and 4x-a, x≥2

A.) Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8. You mus demonstrate all limits.

Homework Equations

ax+7, x<2 and 4x-a, x≥2

The Attempt at a Solution

Approaching from left I got lim x->2- (2a+7) and from right lim x->2+ (8-a)

This is what you mean:
$$ \lim_{x \to 2^-}f(x) = 2a + 7$$
$$ \lim_{x \to 2^+}f(x) = 8 - a$$

Am I trying to find a?

Yes, so that the left-sided limit as x approaches 2 equals 8.

If I set both limits equal I get a = .33, but the question asks determine the value of such that lim x->2- f(x) = 8 ? That would mean a =1/2 because 2* 1/2 = 1 + 7 = 8

What are you saying here? 2 * 1/2 = 1 ≠ 8

, but I don't think that's right unless I'm reading the question wrong..


Edit:
Or do I just state that ax+7 since for this question says lim x->2- (approaching) left f(x) = 8 do I say coming from the left is 8 while from the right it's still 8-a then set those two equal to calculate for "a"? Getting -1 as my answer for "a"?

 

[Torshi]

This is what you mean:
$$ \lim_{x \to 2^-}f(x) = 2a + 7$$
$$ \lim_{x \to 2^+}f(x) = 8 - a$$
Yes, so that the left-sided limit as x approaches 2 equals 8.

Alright, then now I'm assuming set 8 = 8-a

 

[Mark44]

Alright, then now I'm assuming set 8 = 8-a --> which equals a = -1. Thanks

No, for two reasons.

1) You are supposed to

Determine the value of "a" such that lim x->2- (approaching from left) f(x) = 8

2)If 8 = 8 - a, then a = -1 is NOT the solution.

 

[Mark44]

For the left-side limit, a = 1/2.
You had that, but wrote something that didn't make sense.

If a = 1/2, then 2a + 7 = 2(1/2) + 7 = 8

 

[Torshi]

No, for two reasons.

1) You are supposed to
2)If 8 = 8 - a, then a = -1 is NOT the solution.

Sorry, i realized that I switched it to a-8 for some odd reason. Nevermind, I don't know how I came up with that. My fault.

8 = 8 - a
a+8 = 8
a = 0
?

 

[Mark44]

The question is asking about the left-side limit, not the right-side limit.

 

[Torshi]

The question is asking about the left-side limit, not the right-side limit.

Wait, but I thought the question said to determine the value of "a" such that lim x->2- f(x) = 8 meaning that approaching from the left lim x->2- = 8 and from the right that being 8-a, we solve for "a" which is from lim x->2+ (8-a) , so 8 = 8-aif it's the other way around a = 1/2

 

[Torshi]

I think I get it.

2a+7 = 8 then solve for a

a = 1/2

Thanks

 

[Torshi]

For the left-side limit, a = 1/2.
You had that, but wrote something that didn't make sense.

If a = 1/2, then 2a + 7 = 2(1/2) + 7 = 8

I did say that, but I said it without solving so it was just an assumption by looking at the problem, the reason why I didn't go with it because it sounded too easy even though it made sense. My fault. Thank you.