Limit Laws- Constant Times a function: lim[x->a][c*f(x)-g(x)f(x)]/[h(x)]^2

[Barry75]

Hi All,

Many years of taking calculus, I've decided to take it again in preparation of potentially going back for Masters. This is my first post so please be kind. I'm learning about limit laws and I understand that the limit of a constant times a function is equal to the constant times the limit. I also understand the limit law regarding taking of the limit of a quotient (shown in option 2 in screenshot below).

As part of my practice questions, I'm being asked to determine which of 5 (I'm only showing 2) possible answers contains an error in evaluating the limit shown directly below. I'm told to assume that the limit exists and that h(x) is never equal to zero. After getting this question wrong, I was told the answer is option 1 as the scalar cannot be moved outside of the limit because it does not affect the entire limit.

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow a}\, \left\{\dfrac{c\,f(x)\, -\, g(x)\, f(x)}{\left[h(x)\right]^2}\right\}\)

. . . . .\(\displaystyle \displaystyle \mbox{(1) }\qquad c\, \cdot\, \lim_{x \rightarrow a}\, \left\{\dfrac{f(x)\, -\, g(x)\, f(x)}{\left[h(x)\right]^2}\right\}\)

. . . . .\(\displaystyle \displaystyle \mbox{(1) }\qquad c\, \cdot\, \dfrac{\lim_{x \rightarrow a}\, \left[f(x)\, -\, g(x)\, f(x)\right]}{\lim_{x \rightarrow a}\, \left[h(x)\right]^2}\)

But here is point of my confusion. Based on my understanding of the limit laws, as I mentioned above, the limit of a constant times a function is equal to the constant times the limit. How is this different than what I'm told in the answer? Also, what based on understanding of the limit law regarding taking the limit of a quotient, how is option choices 1 and 2 different?

Thanks in Advance,
Barry

 

[HallsofIvy]

Neither option 1 nor option 2 is correct because \(\displaystyle cf(x)- f(x)g(x)\) is NOT equal to c(f(x)- f(x)g(x)). That has nothing to do with limit rules- it is basic algebra. What you can do is factor out the common "f(x)". \(\displaystyle cf(x)- f(x)g(x)= f(x)(c- g(x))\). Given that the limit of h(x) is not 0, we can "take this apart" as \(\displaystyle \left(\lim_{x\to a} f(x)\right)\frac{\lim_{x\to a}c- g(x)}{\lim_{x\to a} h^2(x)}\).

 

[Barry75]

Neither option 1 nor option 2 is correct because \(\displaystyle cf(x)- f(x)g(x)\) is NOT equal to c(f(x)- f(x)g(x)). That has nothing to do with limit rules- it is basic algebra. What you can do is factor out the common "f(x)". \(\displaystyle cf(x)- f(x)g(x)= f(x)(c- g(x))\). Given that the limit of h(x) is not 0, we can "take this apart" as \(\displaystyle \left(\lim_{x\to a} f(x)\right)\frac{\lim_{x\to a}c- g(x)}{\lim_{x\to a} h^2(x)}\).

Thank you and that makes sense. I picked up on your first point as well but with all the points of confusion, it got lost in my mind... I'll see if I can provide feedback to the questions.

Forgetting about the original equation, is there any difference between options 1 and 2? To me, they are equal to one another.

Thanks Again!
Barry

 

[Dr.Peterson]

As part of my practice questions, I'm being asked to determine which of 5 (I'm only showing 2) possible answers contains an error in evaluating the limit shown directly below. I'm told to assume that the limit exists and that h(x) is never equal to zero. After getting this question wrong, I was told the answer is option 1 as the scalar cannot be moved outside of the limit because it does not affect the entire limit.

. . . . .\(\displaystyle \displaystyle \lim_{x \rightarrow a}\, \left\{\dfrac{c\,f(x)\, -\, g(x)\, f(x)}{\left[h(x)\right]^2}\right\}\)

. . . . .\(\displaystyle \displaystyle \mbox{(1) }\qquad c\, \cdot\, \lim_{x \rightarrow a}\, \left\{\dfrac{f(x)\, -\, g(x)\, f(x)}{\left[h(x)\right]^2}\right\}\)

. . . . .\(\displaystyle \displaystyle \mbox{(1) }\qquad c\, \cdot\, \dfrac{\lim_{x \rightarrow a}\, \left[f(x)\, -\, g(x)\, f(x)\right]}{\lim_{x \rightarrow a}\, \left[h(x)\right]^2}\)

But here is point of my confusion. Based on my understanding of the limit laws, as I mentioned above, the limit of a constant times a function is equal to the constant times the limit. How is this different than what I'm told in the answer? Also, what based on understanding of the limit law regarding taking the limit of a quotient, how is option choices 1 and 2 different?

As I understand it, this problem doesn't give you five separate expressions, all but one of which are supposed to equal the given one; rather, they are a sequence of five steps in successively rewriting the expression, and one step is wrong (so that none of the following expressions will be equal to the original, as they would be equal to the wrong step).

You were told that the error is in (1) "as the scalar cannot be moved outside of the limit because it does not affect the entire limit". That's not the best way to say it, but does reflect the error, which as Halls said is not really about the limit itself, but about the algebra. The scalar is not a factor of the entire expression in the limit, so it can't be factored out of the expression, and therefore can't be pulled out of the limit either. That is, you don't have "the limit of a constant times a function" in the first place. The c is only in one term.

If step (1) were valid, then step (2) would correctly follow from it.

 

[Barry75]

As I understand it, this problem doesn't give you five separate expressions, all but one of which are supposed to equal the given one; rather, they are a sequence of five steps in successively rewriting the expression, and one step is wrong (so that none of the following expressions will be equal to the original, as they would be equal to the wrong step).

You were told that the error is in (1) "as the scalar cannot be moved outside of the limit because it does not affect the entire limit". That's not the best way to say it, but does reflect the error, which as Halls said is not really about the limit itself, but about the algebra. The scalar is not a factor of the entire expression in the limit, so it can't be factored out of the expression, and therefore can't be pulled out of the limit either. That is, you don't have "the limit of a constant times a function" in the first place. The c is only in one term.

If step (1) were valid, then step (2) would correctly follow from it.

Thanks Dr.Peterson. You are absolutely right. I misread the problem. They were a sequence of five steps. If I read it correctly, it would have made a lot more sense but I do wish they were as clear in the reason Halls and yourself...

I'm sure I'll be back before end of the class but I'll read the question more carefully next time before I post.

Thanks again!