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# Week 6 Problem Set A.
### Reading.
Thomas Chapters 2.4 and 2.5.
Material covered up to and including Monday lecture (10/3) and up to and including this homework is what you will be expected for on the coming exam.
You can do it!
### Using sandwich theorem.
1. If $\sqrt{5-2x^{2}} \le f(x) \le \sqrt{5-x^{2}}$ for $-1\le x \le 1$, find $\displaystyle\lim_{x\to 0}f(x)$.
2. If $2-x^{2}\le g(x) \le 2\cos(x)$ for all $x$, find $\displaystyle\lim_{x\to 0}g(x)$.
3. Suppose it is shown that for all $x\neq 0$, $$
1- \frac{x^{2}}{6}< \frac{x\sin(x)}{2-2\cos(x)} < 1
$$what is the limit $\displaystyle\lim_{x\to 0} \frac{x\sin(x)}{2-2\cos(x)}$? Explain.
4. Suppose $g(x) \le f(x) \le h(x)$ for all $x\neq 2$, and suppose that $$
\lim_{x\to 2} g(x) = \lim_{x\to 2}h(x)=-5
$$ Can we say anything about $g(2)$, $f(2)$, or $h(2)$? Can way say anything about $\displaystyle\lim_{x\to 2} f(x)$?
### Small angle approximation $\displaystyle\lim_{x\to 0} \frac{\sin(x)}{x}=1$.
Find the following limits.
1. $\displaystyle\lim_{x\to 0} \frac{\sin(3x)}{4x} = ?$
2. $\displaystyle\lim_{x\to 0} \frac{\tan(2x)}{x}= ?$
3. $\displaystyle\lim_{x\to 0} \frac{x\csc(2x)}{\cos(5x)}= ?$
4. $\displaystyle\lim_{x\to 0} \frac{x+x\cos(x)}{\sin(x)\cos(x)}= ?$
5. Recall the half-angle formula: $\cos(A)=1-2\sin^{2}\left( \frac{A}{2} \right)$, use it to find the limit $$
\lim_{x\to 0} \frac{1-\cos(x)}{x}
$$and $$
\lim_{x\to 0} \frac{1-\cos(x)}{x^{2}}
$$
6. Find $\displaystyle\lim_{x\to 0} \frac{1-\cos(x)}{\sin(2x)}$
7. Find $\displaystyle\lim_{x\to 0} \frac{\sin(1-\cos(x))}{1-\cos(x)}$
8. Find $\displaystyle\lim_{x\to 0} \frac{\sin(5x)}{\sin(3x)}$
9. Find $\displaystyle\lim_{x\to 0} \frac{\tan(4x)}{\sin(7x)}$
10. Find $\displaystyle\lim_{x\to 0} \frac{\sin(1-\sqrt{x})}{1-x}$
11. Find $\displaystyle\lim_{x\to 1} \frac{\sin(1-\sqrt{x})}{1-x}$
### One-sided limits.
1. Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false? ![[1 teaching/smc-fall-2023-math-7/week-6/---files/week-6A-problems 2023-10-03 18.01.58.excalidraw.svg]]%%[[1 teaching/smc-fall-2023-math-7/week-6/---files/week-6A-problems 2023-10-03 18.01.58.excalidraw|🖋 Edit in Excalidraw]], and the [[smc-fall-2023-math-7/week-6/---files/week-6A-problems 2023-10-03 18.01.58.excalidraw.dark.svg|dark exported image]]%%
1. $\displaystyle\lim_{x\to -1^{+}}f(x)=1$
2. $\displaystyle\lim_{x\to 0^{-}}f(x)=0$
3. $\displaystyle\lim_{x\to 0^{-}}f(x)=1$
4. $\displaystyle\lim_{x\to 0^{-}}f(x)=\lim_{x\to 0^{+}}f(x)$
5. $\displaystyle\lim_{x\to 0}f(x)$ exists
6. $f(0)$ exists
7. $f$ is continuous at $x=0$
8. $\displaystyle\lim_{x\to 1} f(x)$ exists
9. $\displaystyle\lim_{x\to 1^{-}} f(x)$ exists
10. $\displaystyle\lim_{x\to 1^{+}} f(x)$ exists
11. $f(1)$ exists
### Continuity and continuous extensions.
1. Consider the function $\displaystyle g(x)= \frac{x^{2}-9}{x-3}$ which is not defined at $g(3)$. Choose a value for $g(3)$ so that $g$ becomes continuous at $x=3$.
2. Define $f(1)$ in a way that extends $\displaystyle f(x)= \frac{x^{3}-1}{x^{2}-1}$ becomes continuous at $x=1$.
3. For what value of $A$ is the function $$
f(x)= \begin{cases}
x^{2}-1 & \text{if } x < 3 \\
2Ax & \text{if } x \ge 3
\end{cases}
$$ continuous at every $x$? Hint: Focus on the potentially problematic point.
4. For what values of $A$ is the function $$
f(x) = \begin{cases}
A^{2} x-2A & \text{if } x\ge 2 \\
12 & \text{if } x < 2
\end{cases}
$$continuous at every $x$? Find all such values $A$.
5. For what value of $B$ is the function $$
g(x) = \begin{cases}
\displaystyle\frac{x-B}{B+1} & \text{if } x < 0 \\
x^{2} + B & \text{if } x \ge 0
\end{cases}
$$ continuous at every $x$?
6. For what values of $A$ and $B$ is the function $$
f(x)=\begin{cases}
-2x & \text{if } x \le -1 \\
Ax + B & \text{if } -1 < x < 1 \\
3A x & \text{if } x \ge 1
\end{cases}
$$ continuous at every $x$? Hint: There are two problem points to check for continuity. Write down all the relations you need in order for $f$ to be continuous at each of those problem points. Now, using your relations, solve for $A$ and $B$.
7. For what values of $A$ and $B$ is the function $$
f(x) = \begin{cases}
Ax + B & \text{if } x\le 0 \\
x^{2} + 3A - B & \text{if } 0 < x \le 2 \\
3x - 5 & \text{if }x > 2
\end{cases}
$$ continuous at every $x$?
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