Math $151-$ Workshop $4($limits involving infinity, asymptotes$)$

In class, we learned the following special limits:

$\underset{0}{lim}\frac{sin}{}=1$

$\underset{0}{lim}cos=1$

We also learned that these rules apply for any $$ that approaches $0$ : e$.$g$.,\underset{x0}{lim}\frac{sin57x}{57x}=1.$ Use these facts to find the limits in parts $($a$)-($c$)$ and to answer the question in part $($d$).($If you have heard of something called l$\text{'}$H$$pital$\text{'}$s Rule, you may not use it$.$ Use algebraic manipulation, trig formulas, and limit laws.$)$

$($a$)\underset{x0}{lim}\frac{sin5x}{sinx}.($Hint: start by multiplying by $\frac{5x}{5x}.)$

$($b$)\underset{x0}{lim}\frac{sin5x}{tan3x}.($Hint: $tan=s\frac{in}{cos}.)$

$($c$)\underset{x0}{lim}\frac{1-cos3x}{x^2}.($Hint: What to do with that $1-$cos $3x?$ Maybe use conjugates? Or maybe if you rearrange one of the half$-$angle formulas from Week $1$ you get $2si{n}^2=1-cos2$; could that help? Actually, either of these ideas will work, so choose one!$)$

$($d$)$ Let $f(x)=\{\begin{array}{cc}\frac{sinax}{3x},& x<mn>0</mn>\\ 5,& x0\end{array}.$ Find a such that $f(x)$ is continuous at $x=0.$