$(20$ points$)$

$($a$)$ Find the following limits:

i$.$

$\underset{x0}{lim}\frac{sin3x}{sin5x}$

ii$.$

$\underset{x0}{lim}\frac{e^{3x}-1}{x(x^2-4)}$

$($b$)$ State $($but do not prove$)$ the epsilon$-$delta definition of $\underset{x0}{lim}sin(5x)=1$

$2.(20$ points$)$ Use Intermeliate value theorem to show that $e^x=4-x^3$ has in solution in $|0,2|.$

$3.(20$ points$)$

$($a$)$ Is there any relationship between the slope of the tangent line to a curve $f$ at a point a and the derivative $f^{\text{'}}(a)?($Assuming the derivative of $f$ is defined at that point$).$

$($b$)$ Consider the function $f(x)=3x^2-2x1.$ Find the equation of the tangent line to the curve at the point $x=-1.$

$4.(30$ points$)$ Suppose $x,y$ are differentiable functions of $t,$ and $x^2{y}^2-3y=0.$ Use Implicit differentiation to find $\frac{dy}{ds}.$ No need to simplify the answer.

$5.(10$ points$)$ Using Bisection method, approximate $2^{\frac{1}{3}}$ with an error less than $\frac{1}{8}.$

That's all! Best of Luck:$)$