Calculus help on limits (epsilon delta)

Question

I know the definition and how to prove some limits, but this is a monstrosity. I did some writing but I am still confused. I cannot express for n here because I cannot algebraically manipulate the expression. Please help and please if you can explain easily, it's my third week of freshman year :)

My Attempt:

Answer 1

Hint for (a): the numerator is always between $1$ and $5$ (why?). So you just need to choose $N$ large enough so that for all $n \ge N$ we have $\frac{5}{4n+\sqrt{n}} < \epsilon$.

Hint for (b): $\frac{9n-2}{6n-1}-\frac{3}{2}= - \frac{1}{12n-2}$.

Answer 2

hint

For the first, use the fact that

$$|sin(X)|\le 1$$ and $$n\ge \sqrt{n}$$ to get

$$|u_n|\le \frac{5}{5\sqrt{n}}$$ and look for $N$ such that

$$n>N \implies \frac{1}{\sqrt{n}}<\epsilon$$ or $$n> \frac{1}{\epsilon^2}$$