A cooling chamber is measured over time. Based on the table, what does the limit suggest about its temperature?

Water pressure is recorded near a valve at position $x=2$ meters. What conclusion is best supported by the table?

A solution's concentration is sampled as mixing continues. Which interpretation matches the table's end behavior?

A channel's depth depends on position $x$. What does the table indicate about ${\lim }_{x\to 3}d(x)$?

A force sensor is sampled near $x=4$ meters. Which claim is supported by the table?

Use the graph of $f$. At which value of $x$ does $f$ have a removable discontinuity?

Use the graph of $f$. What type of discontinuity occurs at $x=0$?

Use the graph of $f$. What is ${\lim }_{x\to 2^{-}}f(x)$?

Use the graph of $f$. What is ${\lim }_{x\to 2}f(x)$?

Use the graph of $f$. What is the right-hand limit ${\lim }_{x\to -4^{+}}f(x)$?

Use the graph of $f$. What is $f(-2)$?

A drug concentration is modeled by $C(t)=\frac{80t}{t^2+16}$ mg/L after $t$ hours. What is ${\lim }_{t\to \infty }C(t)$, and what does it mean?

The pressure in a vessel is $P(t)=\frac{3t^2+1}{t^2+4}$ kPa after $t$ minutes. What is its long-term pressure?

A particle's average velocity from $t=0$ to $t=x$ seconds is $\frac{\sqrt{9+x}-3}{x}$ m/s. As $x\to 0$, what does this approach?

The temperature of a reactor is $T(t)=\frac{12t+5}{2t+1}$ °C after $t$ minutes. What does ${\lim }_{t\to \infty }T(t)=6$ mean?

A student writes: “For $f(x)=\frac{x^2-1}{x-1}$, ${\lim }_{x\to 1}f(x)=2$, so $f$ is continuous at 1.” Is the justification valid?

A student writes: “$g(x)=x^3-x-1$ is continuous on $[1,2]$, and $g(1)=-1$ while $g(2)=5$. By the IVT, some $c\in (1,2)$ satisfies $g(c)=0$.” Is this valid?

A student writes: “For $f(x)=|x|$, the left-hand limit at 0 is 0 and the right-hand limit at 0 is 0. Therefore ${\lim }_{x\to 0}f(x)=0$.” Is the justification valid?

A student applies the Mean Value Theorem to $f(x)=\frac{1}{x}$ on $[-1,1]$ and claims a suitable $c$ must exist. Is the justification valid?

For all $x$ near 0, $-x^2\le h(x)\le x^2$. A student concludes that ${\lim }_{x\to 0}h(x)=0$ by the Squeeze Theorem. Is the justification valid?

A shipping charge, in dollars, for mass $m$ kilograms is $C(m)=\{\begin{array}{ll}4m+8& 05\end{array}$. For what value of $k$ is $C$ continuous at $m=5$?

A water rate, in dollars, is $R(v)=\{\begin{array}{ll}2v+6& 010\end{array}$, where $v$ is volume in liters. What value of $a$ makes $R$ continuous at $v=10$?

A heater's temperature is $H(t)=\{\begin{array}{ll}5t+4& 0\le t\le 2\\ kt-2& t>2\end{array}$ °C, where $t$ is in minutes. What value of $k$ makes $H$ continuous at $t=2$?

A pump's flow rate is $Q(t)=\{\begin{array}{ll}{t}^2+3& 0\le t\le 4\\ bt+7& t>4\end{array}$ L/min. What value of $b$ makes $Q$ continuous at $t=4$?

A function satisfies ${\lim }_{x\to 2^{-}}f(x)=6$, ${\lim }_{x\to 2^{+}}f(x)=6$, and $f(2)=1$. Consider: I. ${\lim }_{x\to 2}f(x)=6$. II. $f$ is continuous at 2. III. $f$ has a removable discontinuity at 2. Which statements are true?

A function satisfies ${\lim }_{x\to -1^{-}}f(x)=-2$, ${\lim }_{x\to -1^{+}}f(x)=-2$, and $f(-1)=-2$. Consider: I. The limit exists. II. $f$ is continuous at $-1$. III. $f$ has a removable discontinuity at $-1$. Which statements are true?

A function has a vertical asymptote at $x=3$. Consider: I. ${\lim }_{x\to 3}f(x)$ exists as a real number. II. $f$ is continuous at 3. III. At least one one-sided limit is infinite. Which statement must be true?

Without a calculator, evaluate ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$.

Without a calculator, evaluate ${\lim }_{x\to \infty }\frac{5x^2-1}{2x^2+7}$.

Without a calculator, evaluate ${\lim }_{x\to 0}\frac{\sin (4x)}{x}$.