Write an expression for the limiting temperature by evaluating $\displaystyle\lim_{t\to\infty}T(t)$.

Interpret this limit in the context of the oven.

Calculate the first time at which the oven temperature reaches $150^\circ\mathrm{C}$. Show the equation solved.

Calculate the average rate of change of the temperature from $t=10$ to $t=20$. Include units.

Determine whether the oven temperature at $t=30$ is within $2^\circ\mathrm{C}$ of its limiting temperature. Justify your answer with a calculation.

Estimate the earliest time after which the temperature remains above $170^\circ\mathrm{C}$.

Represent $\displaystyle\lim_{x\to c^-}h_c(x)$ and $\displaystyle\lim_{x\to c^+}h_c(x)$ in terms of $f(c)$ and $g(c)$.

Determine the condition that must be satisfied for $h_c$ to be continuous at $x=c$.

Calculate all values of $c$ in $(-4,5)$ for which $h_c$ is continuous. Show the equation solved.

For $c=2.5$, determine whether $\displaystyle\lim_{x\to c}h_c(x)$ exists. Justify your answer numerically.

Explain why every solution obtained in part (b) produces continuity at $x=c$, rather than merely equal one-sided estimates from the displayed graph.

Verify that

$-x^2\leq v(x)\leq x^2$

for every $x\neq0$.

Evaluate $\displaystyle\lim_{x\to0}(-x^2)$ and $\displaystyle\lim_{x\to0}x^2$.

Justify the value of $\displaystyle\lim_{x\to0}v(x)$ using an appropriate theorem.

Define

$p(x)=\begin{cases}v(x),&x\neq0,\k,&x=0.\end{cases}$

Determine the value of $k$ that makes $p$ continuous at $x=0$.

Verify that the three conditions for continuity at $x=0$ are satisfied for this value of $k$.

Estimate $\displaystyle\lim_{t\to2^-}R(t)$ and $\displaystyle\lim_{t\to2^+}R(t)$.

Determine the type of discontinuity at $t=2$. Justify your classification.

Estimate $\displaystyle\lim_{t\to5}R(t)$.

Determine the value that should replace $R(5)$ to remove the discontinuity.

Verify that the redefined function is continuous at $t=5$.

After the repair in part (b), determine the maximal intervals on which the sensor function is continuous.

Justify that there is at least one time $c$ in $(3,4.5)$ at which $R(c)=2$.

Calculate $\displaystyle\lim_{x\to2}\left(2f(x)-3g(x)\right)$.

Calculate $\displaystyle\lim_{x\to2}\frac{f(x)}{g(x)}$ and verify that the quotient limit law applies.

Define

$F(x)=\begin{cases}f(x),&x\neq2,\k,&x=2.\end{cases}$ Determine the value of $k$ that makes $F$ continuous at $x=2$.

Determine whether

$\displaystyle\lim_{x\to2}\frac{f(x)}{g(x)}=\frac{f(2)}{g(2)}$.

Calculate both quantities and explain your conclusion.

Determine $\displaystyle\lim_{x\to1^-}h(x)$ and $\displaystyle\lim_{x\to1^+}h(x)$.

Evaluate $\displaystyle\lim_{x\to1}h(x)$.

Determine the vertical asymptote of $h$. Justify your answer using the behaviour of $f$.

Determine the horizontal asymptote of $h$. Justify your answer using limits at infinity.

Represent the intervals on which $h$ is continuous.

Determine the value that could be assigned to $h(1)$ to make $h$ continuous at $x=1$.

Verify the hypotheses of the Intermediate Value Theorem on $[-3,-1]$ and justify that there is at least one $c\in(-3,-1)$ for which $f(c)=0$.

Determine whether the Intermediate Value Theorem can be used on $[0,2]$ to guarantee a solution to $f(x)=2$. Explain your answer.

Justify that there is at least one $c\in(2,4)$ for which $f(c)=1$.

Explain why the Intermediate Value Theorem alone does not prove that the solution in part (c) is unique.