Find the exact length of the curve $y=2\sqrt{3}{x}^{\frac{3}{2}}$ for $0\le x\le 4$.

Find the exact length of the curve $y=x-x^2+{\sin -1}^x$ from $x=0$ to $x=1$.

Given that $\underset{x\to 1}{lim}f\left(x\right)=1$, find the largest value of $\epsilon >0$ such that if $|x-1|<\delta$, then $\left|f\right(x)-1|<0.2$. Which of the following values of $\delta$ satisfies this condition?

Given the function $f\left(x\right)=x$, find the largest value of $\delta$ such that if $|x-4|<\delta |$, then $|x-2|<0.4|$.

Which of the following is the correct algebraic method to find the limit $\underset{x\to 2}{lim}\frac{x^2-4}{x-2}$?

Find the arc length of the graph of the function $y=\frac{2}{3}{x}^{\frac{3}{2}}+\frac{1}{3}$ over the interval $0\le x\le 9$.

Find the limit: $\underset{t\to 7}{lim}\frac{7+t^2}{7-t^2}$

Given the function $f\left(x\right)$ defined as follows:

$f\left(x\right)=\left\{\begin{array}{ll}2x+1& ,x<1\\ 3& ,x=1\\ x^2& ,x>1\end{array}\right.$

Find $\underset{x\to 1}{lim}f\left(x\right)$. (If an answer does not exist, enter 'dne.')

Find the limit: $\underset{t\to 5}{lim}\frac{5+t^2}{5-t^2}$