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# Week 5 Problem Set B.
Due: Thursday 10/5
### Reading.
Thomas Chapters 2.2, 2.3, 2.4 for today. And we will finish the rest of chapter 2 between this week and next week.
Stewart Chapters 1.4, to 1.8.
Read about the sandwich theorem in the text for now, I will discuss it later.
There will be another homework set for this week.
### Limits from graphs.
1. Consider the graph of $g(x)$ shown below, find the following limits or explain why they do not exist. ![[1 teaching/smc-fall-2023-math-7/week-5/---files/Pasted image 20230926112447.png]]
1. $\displaystyle \lim_{x\to1}g(x)$
2. $\displaystyle \lim_{x\to 2}g(x)$
3. $\displaystyle \lim_{x\to 3}g(x)$
4. $\displaystyle \lim_{x\to 2.5}g(x)$
2. Which of the following statements about the function $f(x)$ whose graph shown below are true, and which ones are false?![[1 teaching/smc-fall-2023-math-7/week-5/---files/Pasted image 20230926112634.png]]
1. $\displaystyle\lim_{x\to0} f(x)$ exists.
2. $\displaystyle\lim_{x\to0} f(x) = 0$.
3. $\displaystyle\lim_{x\to0} f(x) = 1$.
4. $\displaystyle\lim_{x\to 1} f(x) = 0$.
5. $\displaystyle\lim_{x\to 1} f(x) = 1$. // Typo corrected. //
6. $\displaystyle\lim_{x\to x_{0}} f(x)$ exists at every point $x_{0}$ in $(-1,1)$
7. $\displaystyle\lim_{x\to 1} f(x)$ does not exist.
### Existence of limits.
1. Explain why the limit $\displaystyle \lim_{x\to0} \frac{|x|}{x}$ does not exist. Hint: Think about what happens if $x$ is positive, and if $x$ is negative, and what the graph of $\frac{|x|}{x}$ looks like.
2. Explain why the limit $\displaystyle \lim_{x\to1} \frac{1}{x-1}$ does not exist. Hint: Think about its graph.
3. Suppose a function $f(x)$ is defined at every point $x$ in the interval $[-1,1]$. Does this necessarily imply $\displaystyle \lim_{x\to0}f(x)$ exist? Give reasons to your answer.
4. Suppose a function $f(x)$ is such that $\displaystyle\lim_{x\to1}f(x)$ exists. Does this necessarily imply $f(1)$ is defined? Give reasons to your answer.
5. Give three examples of situations where a function $f(x)$ whose limit would not exist at some point $x=a$. You can use a sketch and describe these situations.
### Calculating limits.
Find the limits in the following. Some you may need algebra to resolve the "dividing by zero" issue:
1. $\displaystyle\lim_{x\to-7}(2x+5)$
2. $\displaystyle\lim_{t\to6} 8(t-5)(t-7)$
3. $\displaystyle\lim_{x\to 2} \frac{x+3}{x+6}$
4. $\displaystyle\lim_{x\to-1}3(2x-1)^{2}$
5. $\displaystyle\lim_{h\to0} \frac{3}{\sqrt{3h+1}+1}$
6. $\displaystyle\lim_{h\to0} \frac{\sqrt{5h+4}-2}{h}$ (Hint: Use the conjugate)
7. $\displaystyle \lim_{x\to 5} \frac{x-5}{x^{2}-25}$
8. $\displaystyle \lim_{x\to -5} \frac{x^{2}+3x-10}{x+5}$
9. $\displaystyle \lim_{x\to 1} \frac{ \frac{1}{x}-1}{x-1}$
10. $\displaystyle \lim_{x\to 1} \frac{x^{4}-1}{x^{3}-1}$ (Try to factor, both the numerator and denominator share a common factor)
11. $\displaystyle \lim_{x\to 9} \frac{\sqrt{x}-3}{x-9}$
12. $\displaystyle \lim_{x\to 1} \frac{x-1}{\sqrt{x+3}-2}$
13. $\displaystyle \lim_{x\to 2} \frac{\sqrt{x^{2}+12}-4}{x-2}$
14. $\displaystyle \lim_{x\to -3} \frac{2-\sqrt{x^{2}-5}}{x+3}$
15. $\displaystyle \lim_{x\to0}(2\sin x -1)$
16. $\lim_{x\to0}\sec(x)$
17. $\displaystyle \lim_{x\to0} \frac{1+ x + \sin(x)}{3\cos x}$
18. $\displaystyle\lim_{x\to -\pi}\sqrt{x+4}\cos(x+\pi)$
### Using limit rules.
1. Suppose $\displaystyle\lim_{x\to 4} f(x) = 0$ and $\displaystyle\lim_{x\to 4}g(x)=-3$. Find:
1. $\displaystyle\lim_{x\to4}(g(x)+3)$
2. $\displaystyle\lim_{x\to4}xf(x)$
3. $\displaystyle\lim_{x\to4}(g(x))^{2}$
4. $\displaystyle\lim_{x\to 4} \frac{g(x)}{f(x)-1}$
2. Suppose $\displaystyle\lim_{x\to b}f(x)=7$ and $\displaystyle\lim_{x\to b}g(x)=-3$. Find:
1. $\displaystyle\lim_{x\to b}(f(x)+g(x))$
2. $\displaystyle\lim_{x\to b}f(x)\cdot g(x)$
3. $\displaystyle\lim_{x\to b}4 g(x)$
4. $\displaystyle\lim_{x\to b} \frac{f(x)}{g(x)}$
3. Suppose that $\displaystyle\lim_{x\to -2}p(x)=4$, $\displaystyle\lim_{x\to -2} r(x)=0$, and $\displaystyle\lim_{x\to-2}s(x)=-3$. Find:
1. $\displaystyle\lim_{x\to-2}(p(x)+r(x)+s(s))$
2. $\displaystyle\lim_{x\to-2}(p(x)\cdot r(x) \cdot s(x))$
3. $\displaystyle\lim_{x\to-2} \frac{-4p(x)+5 r(x)}{s(x)}$
### Limits of average rates of changes.
In each of the following, evaluate the limit $$
\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
$$for the specified function $f(x)$ and the specified point $x$. This is the slope of the tangent line at $(x,f(x))$ on the graph of $f$:
1. $f(x)=x^{2}$, at $x=-2$
2. $\displaystyle f(x)=\frac{1}{x}$, at $x=-2$
3. $f(x)=\sqrt{3x+1}$, at $x=0$
### Theory and practice.
1. If $\displaystyle\lim_{x\to4} \frac{f(x)-5}{x-2}=1$, find $\displaystyle\lim_{x\to 4} f(x)$.
2. If $\displaystyle\lim_{x\to -2} \frac{f(x)}{x^{2}}=1$, find $\displaystyle\lim_{x\to-2}f(x)$ and $\displaystyle\lim_{x\to-2} \frac{f(x)}{x}$.
3. If $\displaystyle\lim_{x\to 2} \frac{f(x)-5}{x-2}=3$, find $\displaystyle \lim_{x\to 2}f(x)$.
4. If $\displaystyle\lim_{x\to 2} \frac{f(x)-5}{x-2}=4$, find $\displaystyle \lim_{x\to 2}f(x)$.
5. If $\displaystyle\lim_{x\to 0} \frac{f(x)}{x^{2}}=1$, find $\displaystyle\lim_{x\to 0}f(x)$ and $\displaystyle\lim_{x\to 0} \frac{f(x)}{x}$.
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