I am teaching a real analysis class. Students in the class have inconsistent high school algebra skills. They now have a complete but tenuous understanding of $\varepsilon$-$\delta$ limits. I want to give them problems on which to consolidate their understanding of this definition and how to prove that a given proposed limit satisfies it. I want more examples to try out with them that are interesting but do not overly tax their algebra skills or demand too much sophisticated inequality-based reasoning, which is entirely new to them. I am looking for problems. Answers to this question could consist of problem suggestions, or pointing me to sources.
To calibrate what I'm looking for, I will go in detail through an example I tried with them last class that was too hard. (It is not very hard if you have certain analysis skills, but these are precisely the skills they don't have.)
We worked through an example like $\lim_{x\to -5}10x$; they felt they understood everything that was going on. I threw $\lim_{x\to 4} 3x^2$ at them, thinking it was just one level up, but it was way too much. They attacked it like this:
The limit is 48, by substitution. We need to find the constraint on $x$ that causes $|3x^2 - 48|$ to be $<\varepsilon$. So we need
$$-\varepsilon < 3x^2 - 48 < \varepsilon$$ i.e. $$-\varepsilon + 48 < 3x^2 < \varepsilon + 48$$ i.e. $$ -\frac{\varepsilon}{3} + 16 < x^2 < \frac{\varepsilon}{3} + 16$$ and so $$ \sqrt{-\frac{\varepsilon}{3} + 16} < x < \sqrt{\frac{\varepsilon}{3} + 16}.$$
Well and good (subject to $\varepsilon \leq$ something), but now they needed a $\delta$ such that $0<|x-4|<\delta$ will force this last inequality. Using a taylor series approximation for the square root is not something they would have ever thought of. I proposed working with the previous inequality prior to the square root. My idea was to try to get them to square $4-\delta < x < 4 + \delta$ and then choose $\delta$ to force $-\varepsilon / 3 + 16 < (4-\delta)^2 < x^2 < (4+\delta)^2 < \varepsilon / 3 + 16$. First of all, I had to hand them this move. Secondly, finding the necessary $\delta$ was still hard, once I had handed them this setup. They went:
$$-\frac{\varepsilon}{3} + 16 < (4 - \delta)^2 = \delta^2 - 8\delta + 16$$ so we need $$ -\frac{\varepsilon}{3} < \delta^2 - 8\delta$$ or $$ 8\delta - \delta^2 < \frac{\varepsilon}{3}$$
and at this point started trying to remember how to solve a quadratic equation. I stopped them and pointed out that $\delta^2$ is positive, therefore this inequality is guaranteed as soon as $8\delta < \varepsilon / 3$, and they immediately concluded that $\delta$ had to be $<\varepsilon / 24$.
Then we looked at the right inequality, $(4 + \delta)^2 < \varepsilon/3 + 16$. They got as far as
$$\delta^2 + 8\delta < \frac{\varepsilon}{3},$$
and again I had to hand them a trick: pointing out that it never hurts to take a smaller delta, we can assume $\delta \leq 1$, implying $\delta^2 \leq \delta$, so that $\delta^2 + 8\delta \leq 9\delta$. From here, they concluded we need $\delta < \varepsilon / 27$. So in the end they chose $\delta =\min(1,\delta / 28)$.
Because of all the tricks I had to hand them, and handling the sides separately, it was hard for them to see the big picture at once, so in the end, I don't feel the problem consolidated their understanding of the definition.
What I want is several problems that are not all the same (i.e. not just a bunch of $\lim_{x\to c} f(x)$ for $f(x)$ linear) but that do not put up the types of algebraic/analytic roadblocks I described above: places where you either have to (a) do some slightly heavier HS algebra, like solve a quadratic and then reason clearly about how its solution plays into these inequalities, or (b) use the analyst's trick of exploiting the slack afforded by the inequalities, as above when I pointed out to them we could take $\delta \leq 1$ in order to have $\delta^2 \leq \delta$, which is its own separate skill I would prefer to work on with them once we already have the essential goals of the $\varepsilon$-$\delta$ proof mastered.
Again, I am asking either for your own suggested problems, or suggestions about good sources for such problems. Thanks in advance.
ADDENDUM: In addition to Brendan's nice suggestion of degree 1 rational functions, I found some other nice ideas in Exploratory Examples for Real Analysis, by Snow and Weller:
Piecewise-linear functions, where you're taking the limit at a joint.
Things like $(x-5)\sin x$ as $x\to 5$, where the key is just to notice that $|\sin x|\leq 1$.
I haven't tried them with my students yet to see how they will work, but these are just the types of ideas I was looking for.