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I got this problem i've tried to solve but i don't know how to proceed. $$\begin{cases} 3\sin(4x)&&\text{if }x\leq 0 \\ mx+q&&\text{if }x>0 \end{cases}$$ Find the value of $m$ and $q$ so that the function is continuos in R.

I've tried this $$\lim_{x\to 0^-} 3\sin(4x) = \lim_{x\to 0^+}mx+q$$ So that the $q$ value equals to $0$ but I don't know.

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    $\begingroup$ "the $q$ value equals $0$" looks like a good conclusion. You might want to consider whether the function would then be continuous for any value of $m$ $\endgroup$
    – Henry
    Commented Nov 7, 2022 at 16:33
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    $\begingroup$ Hi! I've taken the step of editing your MathJax. Note that (as with regular LaTeX), MathJax treats any letters as variables, unless they are part of a control sequence (or are enclosed in a \text{} command)). $\endgroup$
    – Brian Tung
    Commented Nov 7, 2022 at 16:37
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    $\begingroup$ Further comments, if you want to make these changes: $1$. It is generally not considered obligatory to use parentheses in $\sin 4x$, even though it's technically less ambiguous that way. $2$. The reals are often denoted either $\mathbb{R}$ ($\mathbb{R}$) or, more traditionally, $\mathbf{R}$ ($\mathbf{R}$). $\endgroup$
    – Brian Tung
    Commented Nov 7, 2022 at 16:42
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    $\begingroup$ @BlobFish If my answer was satisfactory in resolving your question, can you please vote for it and/or click the checkmark to accept it? $\endgroup$
    – Nico
    Commented May 23, 2023 at 22:35

1 Answer 1

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You are correct in determining that $q = 0$.

Since we are taking the limit at $x = 0$, and anything times $0$ is still $0$, the coefficient of $x$ does not have any effect on the answer. Therefore, $m$ can be any real number, and $q = 0$ (as mentioned above). We can represent this answer as $\boxed{q = 0, m \in \mathbb{R}}$ or $\boxed{q = 0, m \in (-\infty, \infty)}$.

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