Problems $1-7$ regard indeterminate forms $($see section $6\mathrm{.}8)$ and problems $8-12$ regard integration by parts $($see section $7\mathrm{.}1).$

Find the limits in problems $1-6.$ Use l'Hopital's Rule where appropriate. If there is a more elementary method, consider using it$.$ If l'Hopital's Rule does not apply, explain why.

$\underset{t0}{lim}(\frac{e^{3t}-1}{t})$

$\underset{x{0}^{}}{lim}\frac{lnx}{\sqrt[\phantom{2}]{x}}$

$\underset{x{0}^{}}{lim}(sinx*lnx)$

$\underset{x}{lim}(x{e}^{(\frac{1}{x})}-x)$

$\underset{x{0}^{}}{lim}{x}^{(x^2)}$

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

The figure shows the sector of a circle with central angle $.$ Let $A()$ be the area of the segment between the chord $PR$ and the arc $PR.$ Let $B()$ be the area of the triangle $PQR.$ Find $\underset{0}{lim}\frac{A()}{B()}.$

Evaluate the integrals in problems $8-11.$

$8.{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x*cos(5x)dx.$

$9.{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x*2^{x}dx.$

$10.{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cos\sqrt[\phantom{2}]{x}dx.$

Hint: First make the substitution $t=\sqrt[\phantom{2}]{x}$ and then use integration by parts. I need problems number $3,5,7,9,10$ ONLY