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# Week 5 Problem Set C.
Due: Thursday 10/5
Now we do real math.
[[1 teaching/smc-fall-2023-math-7/week-5/formal-definition-of-limits|Here are some additional notes and examples on proving limits to supplement what we discussed in class.]]
You can do it! If you can do this, I am beyond proud of you!
(You will want to know how to do this, say by this coming Thursday...for perhaps obvious reasons.)
### Formal definition of limits. (Or, precise definition of limits)
A brief "review".
Recall the formal definition of limits, which is given as follows:
> **Definition.**
> We say $\displaystyle\lim_{x\to a}f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $0 < |x - a| <\delta$, we have $|f(x)-L| < \epsilon$.
Graphically, if we say $\displaystyle\lim_{x\to a} f(x) = L$, this means for any $\epsilon$-tube we draw around $y=L$, we can always choose a $\delta$-window around $x=a$ (which are the intervals $(a-\delta,a)$ and $(a,a+\delta)$), such that the function $f(x)$ above the $\delta$-window will lie inside the $\epsilon$-tube:
![[1 teaching/smc-fall-2023-math-7/week-5/---files/week-5C-problems 2023-09-28 15.57.55.excalidraw.svg]]
%%[[1 teaching/smc-fall-2023-math-7/week-5/---files/week-5C-problems 2023-09-28 15.57.55.excalidraw|🖋 Edit in Excalidraw]], and the [[smc-fall-2023-math-7/week-5/---files/week-5C-problems 2023-09-28 15.57.55.excalidraw.dark.svg|dark exported image]]%%
A proof of $\displaystyle\lim_{x\to a}f(x) = L$ should have the following structure (in order to satisfy the definition):
>For $\epsilon > 0$, take $\delta =$ ........... .
>Then whenever $0 < |x-a|<\delta$,
>.
>.
>.
>.
>.
>We get $|f(x)-L| < \epsilon$. $\blacksquare$
We will need to do some work to figure out what to pick for $\delta$. For "simple cases" (polynomial, rational functions, and roots), we can reverse engineer by manipulating $|f(x)-L| < \epsilon$, and with algebra extract some information about $|x-a|$. You may need to control the size of some terms as shown in class.
For each of the following, prove the proposed limits using the definition of limits as shown in class.
1. Prove $\displaystyle\lim_{x\to 3}(5x-2)=13$
2. Prove $\displaystyle\lim_{x\to -2}(4-6x)=16$
3. Prove $\displaystyle\lim_{x\to 3} x^{2} = 9$
4. Prove $\displaystyle\lim_{x\to 2} x^{2}+x = 6$
5. Prove $\displaystyle\lim_{x\to 3} \frac{1}{x} = \frac{1}{3}$.
6. Prove $\displaystyle\lim_{x\to 4} \frac{1}{x^{2}+x} = \frac{1}{20}$
7. Prove $\displaystyle\lim_{x\to 5} \frac{x+2}{x-3}= \frac{7}{2}$
8. Prove $\displaystyle\lim_{x\to 2} \sqrt{x+3} = \sqrt{5}$. (Hint: use conjugates)
9. Prove $\displaystyle\lim_{x\to 1} (x^{3}+4x^{2}-3x+1) = 3$
10. Prove $\displaystyle\lim_{x\to 2} \frac{x^{2}+3x}{x+5} = \frac{10}{7}$.
When doing some of these, remember your algebra tool box: Factoring, long division, algebraic conjugates.
(Try to do it without calculators, and no need to divide decimal numbers, just leave them in fractions, if any. Some of the numbers are not "nice" above.)
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