1. Limits and Continuity

How many tangent lines to the curve $y=\frac{x}{x+1}$ pass through the point $(1,\frac{1}{2})$?

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 exact length of the curve given by $x=2\sqrt{{\sqrt{t}}^3}$, $y=t^2-2$ for $0\le t\le 4$.

At what point do the curves $r_1\left(t\right)=(t,2-t,35+t^2)$ and $r_2\left(s\right)=(7-s,s-5,s^2)$ intersect?

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

Evaluate the following limit, if it exists. If the limit does not exist, select 'DNE'.
$$\underset{(x,y)\to (0,0)}{lim}\frac{xy}{x^2+y^2}$$

Given that $d\left(t\right)$ is a known function and $d$ is a specific value, which of the following procedures will find the time $t$ at which $d\left(t\right)=d$?

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

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).