1. Limits and Continuity

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?

Even function limits Suppose f is an even function where lim x→1^− f(x)=5 and lim x→1^+ f(x)=6. Find lim x→−1^− f(x) and limx→−1^+ f(x).

Determine the following limits.

$\underset{z\to 4}{\lim }\frac{z-5}{{(z^2-10z+24)}^2}$

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

Determine the following limits.

$\underset{\theta \to 0^{-}}{\lim }\frac{\sin \theta }{{\cos }^2\theta -1}$

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

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

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.

$\lim_{x\rarr0}x^3+5x^2-7x+3$

Find the limit.

$\lim_{x\rarr2}\sqrt{x^2+5}$

Find the limit.

$\lim_{x\rarr3}\frac{x^2+2x-3}{x-2}$

Find the limit.

$\lim_{x\rarr0}\frac{3x^2+7x}{x}$

Find the limit.

$\lim_{x\rarr2}\frac{x^2-7x+12}{x-3}$

Find the limit.

$\lim_{x\rarr-2}\frac{x^2-5x-14}{x+2}$

Find the limit.

$\lim_{x\rarr-\pi}\frac{\sin x}{x}$