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In extreme details, may you please discuss the Continuity of these two questions I already have the answer in my book, but I don't know how they got it, so from a to z may you explain?

  1. $f(x)= \frac{x-1}{\cos x}$

  2. $f(x)= \frac{x^2+3}{1+\sin x}$

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    Solve? Solve what? Are you asking for the points at which these functions are continuous? – 2017-01-16
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    @MPW sorry I mean '' discuss '' – 2017-01-16
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    You should provide a little more information about what you can do. Did you sketch graphs of these functions? – 2017-01-16
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    @EthanBolker excuse me sir what graphs? More information is not needed, and I need the algebraic solution. (Discuss the Continuity of each function in R) – 2017-01-16
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    If you draw the graphs you will get some intuition about what you want to prove with algebra. That intuition should be a part of any "discussion". Then you should ask here about how you start the algebra. Without that effort on your part you may not get a complete answer - we're here to help you, not to do all the work. I note that you asked essentially the same question about other functions an hour before you asked this one. – 2017-01-16
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    This time we have functions that are continous at every position, where they are defined. The polynomials, the functions $\sin(x)$ and $\cos(x)$ are continous as well as sums, products or quotients of them (Unless the denominator is $0$). You just have to find the positions, where the denominator is $0$, here. At all other positions , we have continuity. – 2017-01-16
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    @Peter that is exactly what I want to know when can $cosX$ =0 and when $1+sinx$ =0??? – 2017-01-16
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    $\cos(x)=0$ for $x=(k+\frac{1}{2})\pi$ , $k\in \mathbb Z$ – 2017-01-16
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    @Peter that is for cosine what about the sine one9 – 2017-01-16
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    $1+\sin(x)=0$ for $x=(2k+\frac{3}{2})\pi$ , $k\in\mathbb Z$ – 2017-01-16

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