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Thread: [College Calculus] Continuity at A?

  1. #1
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    [College Calculus] Continuity at A?

    Hey guys, so i have this problem from my final exam, and I have no clue on how to do and would like to actually understand it (I uploaded the pic of the problem below).

    It is about finding the value of a that makes h(x)g(x). I know that a function is continuous at a point if the limit exists on both sides and that the limit is at the same value. However, I never dealt with product of functions and how to find if there's a value for a that makes the product of functions continuous.

    How would i otherwise prove or disprove that there's a value for a?

    [PIC] https://imgur.com/a/QDnGe
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    Re: [College Calculus] Continuity at A?

    let $f,~g,~h$ all be as described.

    $\displaystyle \lim_{x \to 1^-}(fg)(x) = (1)(-2) = -2$

    $\displaystyle \lim_{x \to 1^+}(fg)(x) = (2)(-1) = -2$

    so clearly we need $h(x)$ defined such that $(hg)(1) = -2$

    and thus we define

    $h(x) = \begin{cases}f(x) &x \neq 1 \\ 2 &x=1 \end{cases}$
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    Re: [College Calculus] Continuity at A?

    Thanks but why is it (hg) (1) not (fg)(1)? How are the functuons related? Kinda confused of why h (x) looks like.
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    Re: [College Calculus] Continuity at A?

    Quote Originally Posted by lc99 View Post
    Thanks but why is it (hg) (1) not (fg)(1)? How are the functions related? Kinda confused of why h (x) looks like.
    $\begin{gathered}
    {\lim _{x \to {1^ - }}}f(x) = 1\,\& \,{\lim _{x \to {1^ + }}}f(x) = 2 \hfill \\
    {\lim _{x \to {1^ - }}}g(x) = - 2\,\& \,{\lim _{x \to {1^ + }}}g(x) = - 1 \hfill \\
    {\lim _{x \to {1^ - }}}fg(x) = - 2\,\& \,{\lim _{x \to {1^ + }}}fg(x) = - 2 \hfill \\
    {\lim _{x \to 1}}h(x) = - 2 \hfill \\
    \end{gathered}$

    Do you understand the above?
    Can you explain each step by giving reasons?
    Thanks from lc99
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    Re: [College Calculus] Continuity at A?

    Yeah, i understand that the limit of one side of a function is gonna be different on another side. I think i understand that h(x) is simply just f(x) rewritten at x=1, and that it's just multiplying with g(x) so that both sides of limit of h(x)g(x) goes to -2 when x=1.
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    Re: [College Calculus] Continuity at A?

    In order for a function to be continuous at a point, it must be defined at a point.

    $f(x)$ is not defined at x=1. $g(x) $ is. Their product is not defined at $x=1$, but the limit as x approaches 1 of their product exists and equals -2. Therefore, their product has a removable discontinuity at x=1. You are asked to find a value that will remove the discontinuity. $h(x) $ looks exactly like $f (x) $ except it IS defined at x=1. If defined so that $h (1)=-2$ then the product $h (x)g (x)=f (x)g (x) $ for all x not equal to 1 and $-2g (1)=-2$ at x=1.
    Thanks from lc99 and topsquark
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    Re: [College Calculus] Continuity at A?

    Thanks a lot! I can visualize it better since I didn't really see where h (x) came from. But now I do
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    Re: [College Calculus] Continuity at A?

    Hey, so i recently found out that a = 2 and not a = -2? I'm kinda confused as to why because the product of those functions from the right of 2 and left of 2 is -2 . But, a is suppose to be 2 according to my teacher !
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    Re: [College Calculus] Continuity at A?

    Quote Originally Posted by lc99 View Post
    Hey, so i recently found out that a = 2 and not a = -2? I'm kinda confused as to why because the product of those functions from the right of 2 and left of 2 is -2 . But, a is suppose to be 2 according to my teacher !
    Unless the original posting is mistaken, then your teacher is mistaken.
    Please check the first post.
    Thanks from topsquark
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    Re: [College Calculus] Continuity at A?

    Hmm. The pic of my problem is of the orginial graphs of f(x) and g (x). I drew the f (x) graph.. could that be wrong?. I was suppose to find a for (hg)(g (x)). Unless he used the wrong answer key, this was a problem on the midterm exam!!
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    Re: [College Calculus] Continuity at A?

    Quote Originally Posted by lc99 View Post
    Hmm. The pic of my problem is of the orginial graphs of f(x) and g (x). I drew the f (x) graph.. could that be wrong?. I was suppose to find a for (hg)(g (x)). Unless he used the wrong answer key, this was a problem on the midterm exam!!
    Is this a professor? A TA? At what level is your instructor? It is time to go over his head.
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    Re: [College Calculus] Continuity at A?

    He's a TA and is a graduate student. I thought he knows his stuff cause he seems like it XD

    In Romsek's post, he noted that at x=1 , a=2?? Though
    Last edited by lc99; Nov 3rd 2017 at 07:26 AM.
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    Re: [College Calculus] Continuity at A?

    Quote Originally Posted by lc99 View Post
    He's a TA and is a graduate student. I thought he knows his stuff cause he seems like it XD

    In Romsek's post, he noted that at x=1 , a=2?? Though
    Ok, I was misunderstanding the confusion. At $x=1$, you have $h(x)g(x) = a(-1) = -a$. So, if you want the function to be continuous, you need $a=2$ so that $h(x)g(x) = -2$ at x=1. So, what is your confusion with this problem? You were given the answer already.
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