Title: Math 2250 Chapter 2 (Limits)

Q: [latex]
The \emph{average rate of change} of $f(x)$ over $[a,b]$ is
[/latex]
A: [latex]
the change in $y$ divided by the change in $x$ or
\begin{equation*} \frac{\Delta y} {\Delta x}
= \frac{f(b) - f(a)} {b - a}.
\end{equation*}
[/latex]

Q: [latex]
The limit of a sum
$$
\lim_{x \rightarrow c} f(x) + g(x) =
$$
[/latex]
A: [latex]
The sum of the limits
$$
\lim_{x \rightarrow c} f(x) + g(x) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x).
$$
[/latex]

Q: [latex]
The limit of a difference
$$
\lim_{x \rightarrow c} f(x) - g(x) =
$$
[/latex]
A: [latex]
The difference of the limits
$$
\lim_{x \rightarrow c} f(x) - g(x) = \lim_{x \rightarrow c} f(x) - \lim_{x \rightarrow c} g(x).
$$
[/latex]

Q: [latex]
If $k$ is a constant, the limit of a constant multiple of a function
$$
\lim_{x \rightarrow c} k \cdot f(x) =
$$
[/latex]
A: [latex]
is that multiple of the limit
$$
\lim_{x \rightarrow c} k \cdot f(x) =
k \lim_{x \rightarrow c} f(x).
$$
[/latex]

Q: [latex]
The limit of a product
$$
\lim_{x \rightarrow c} f(x) \cdot g(x)=
$$
[/latex]
A: [latex]
is the product of the limits
$$
\lim_{x \rightarrow c} f(x) \cdot g(x)=
\left( \lim_{x \rightarrow c} f(x) \right) \cdot \left( \lim_{x \rightarrow c} g(x) \right)
$$
[/latex]

Q: [latex]
The limit of a quotient
\begin{equation*}
\lim_{x \rightarrow c} \frac{f(x)}{g(x)} =
\end{equation*}
[/latex]
A: [latex]
is the quotient of the limits \textbf{if the limit of the denominator is not zero}
\begin{equation*}
\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)} \text{ if } \lim_{x \rightarrow c} g(x) \neq 0.
\end{equation*}
[/latex]

Q: [latex]
The limit of the $n$th-power of a function
$$
\lim_{x \rightarrow c} [f(x)]^n =
$$
[/latex]
A: [latex]
is the $n$th power of the limit of the function
$$
\lim_{x \rightarrow c} [f(x)]^n = \left[\lim_{x \rightarrow c} f(x) \right]^n.
$$
[/latex]

Q: [latex]
The limit of the $n$-th root of a function
\begin{equation*}
\lim_{x \rightarrow c} \root n \of {f(x)} =
\end{equation*}
[/latex]
A: [latex]
is the $n$th root of the limit of the function.
\begin{equation*}
\lim_{x \rightarrow c} \root n \of {f(x)} = \root n \of { \lim_{x \rightarrow c} f(x) }
\end{equation*}
[/latex]

Q: [latex]
We say that \emph{the limit of $f(x)$ as $x \rightarrow c$ is $L$}
or
\begin{equation*}
\lim_{x \rightarrow c} f(x) = L
\end{equation*}
if
[/latex]
A: [latex]
for every $\epsilon > 0$ there is a $\delta > 0$ so that
\begin{equation*}
0 < |x -c| < \delta \implies |f(x) - L| < \epsilon
\end{equation*}
[/latex]

Q: [latex]
The limit of $f(x)$ as $x$ approaches $c$ \emph{from the right}
\begin{equation*}
\lim_{x \rightarrow c^+} f(x)
\end{equation*}
[/latex]
A: [latex]
is the limit
\begin{equation*}
\lim_{x \rightarrow c^+} f(x)  = \lim_{h \rightarrow 0} f(c + |h|)
\end{equation*}
of values of $f(x)$ where $x > c$ and $x$ approaches $c$.
[/latex]

Q: [latex]
The limit of $f(x)$ as $x$ approaches $c$ \emph{from the left}
\begin{equation*}
\lim_{x \rightarrow c^-} f(x) =
\end{equation*}
[/latex]
A: [latex]
is the limit
\begin{equation*}
\lim_{x \rightarrow c^-} f(x) = \lim_{h \rightarrow 0} f(c - |h|)
\end{equation*}
of values of $f(x)$ for $x < c$ as $x$ approaches $c$.
[/latex]

Q: [latex]
The limit
\begin{equation*}
\lim_{x \rightarrow 0} \frac{\sin x}{x} =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1
\end{equation*}
[/latex]

Q: [latex]
A function $f(x)$ is \emph{continuous} at the point $c$ if
[/latex]
A: [latex]
you can get the limit of $f(x)$ as $x \rightarrow c$ by plugging in $c$, or
\begin{equation*}
\lim_{x \rightarrow c} f(x) = f(c)
\end{equation*}
Almost every function you know is continuous \emph{where it is defined}. A discontinuous function is
\begin{equation*}
\operatorname{sgn}(x) = \begin{cases}
         1 & \text{if } x \geq 0 \\
        -1 & \text{if } x < 0 \\
\end{cases}
\end{equation*}
[/latex]

Q: [latex]
If $f(x)$ and $g(x)$ are continuous at $x = c$ then
[/latex]
A: [latex]
\begin{enumerate}
\item $f(x) + g(x)$
\item $f(x) - g(x)$
\item $k \cdot f(x)$
\item $f(x) \cdot g(x)$
\item $f(x) / g(x)$ (if $g(c) \neq 0$)
\item $f(x)^n$
\item $\root n \of {f(x)}$
\end{enumerate}
are all continuous at $x = c$.
[/latex]

Q: [latex]
If $g(x)$ is continuous at $c$ and $f(x)$ is continuous at $g(c)$, then
\begin{equation*}
\lim_{x \rightarrow c} f(g(x)) =
\end{equation*}
[/latex]
A: [latex]
we have
\begin{equation*}
\lim_{x \rightarrow c} f(g(x)) = f\left( \lim_{x \rightarrow c} g(x) \right).
\end{equation*}
[/latex]