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With trying with examples, I found that the function $$f(x) = \frac{ax+b}{cx+d}$$ has a jump continuity when $a\cdot \frac{-d}{c} + b = 0$ at point $(-d/c, \frac{a+b}{c+d})$

However, I could not find a calculation that leads to the result $y_0 = \frac{a+b}{c+d}$. I somehow failed to calculate the limit of $\lim_{x\rightarrow-d/c}f(x)$. What am I missing here?

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    $\begingroup$ When $x=-d/c$, the function $f$ is undefined. $\endgroup$
    – Kenta S
    Commented Sep 29, 2022 at 17:10
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    $\begingroup$ when $ad-bc=0$ the function is constant, graph a line. Otherwise the graph is a hyperbola. $\endgroup$
    – Will Jagy
    Commented Sep 29, 2022 at 17:21
  • $\begingroup$ @WillJagy Yes I understand that. Just I cannot get the $\frac{a+b}{c+d}$ with the calculation $\endgroup$
    – CryForGG
    Commented Sep 29, 2022 at 17:27
  • $\begingroup$ Since it's a horizontal line, it's a constant and you can pick almost any $x$ value to plug in. Technically this won't work when $c+d=0$, since that'd be equivalent to plugging in to make $f(1)$, which is undefined. So you can choose a different value, like $f(0)=\frac b d$ or $\frac a c$. $\endgroup$
    – Merosity
    Commented Sep 29, 2022 at 17:59
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    $\begingroup$ @Merosity Oh, now I understand, just put some values in XD. Thank you! $\endgroup$
    – CryForGG
    Commented Sep 29, 2022 at 21:18

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