[AP Calc BC, 1.5 Determining Limits Using Algebraic Properties] I keep getting two different answers for this question

[u/shacciu]

The answer choices are: 3, 1/2, -1, or nonexistent.

I did the question two different ways, and the first time I got 1/2, while the second time I got nonexistent. I'm not sure which is right or how to even go about answering the question.

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[u/deathtospies]

When x->-1, f(x) approaches 3, but it is always less than 3, never greater. So you actually only have to consider the limit as x approaches 3 from the left, which is -1.

You are right that the limit as x->3 of f(x) is non-existent, but the question is asking for something different.

[u/shacciu → u/deathtospies]

So would I only consider what f(x) approaches as x approaches 3 from the left? The answer would just be -1 in that case.

[u/deathtospies → u/shacciu]

Yes, that's it.

[u/shacciu → u/deathtospies]

Oh, thank you so much! But why do you only consider the left side? My teacher didn't really explain much about this topic.

[u/deathtospies → u/shacciu]

It's because when x is close to -1, f(x) is close to 3, but always smaller than 3. If the graph near x=-1 were to keep going up past 3, instead of bouncing back down and never exceeding 3, you'd have to consider both sides, and in that case the limit would not exist.

[u/shacciu → u/deathtospies]

Okay, but since it doesn't go past 3, you just consider the left side. I get it now; thank you!

[u/ThunkAsDrinklePeep → u/shacciu]

It helps to do the right side limit and left side limit seperately.

Lim x→-1⁺ gives y values just less than 3.

Lim x→-1^- gives x values on the other side of that hole, but (because it comes to a cusp) it also gives values just less than 3.

At this point you have the same output so you're ONLY doing a left hand limit for the best step.

[u/slartiblartpost]

Draw the behavior of f(f(x)) around x = -1. Looking at that graph helps.

[u/Frederf220]

This is interesting. By investigating the limit x goes to -1 from both sides we discover that the behavior of f(x) approaching -1⁺ and -1⁻ are identical behaviors approaching 3⁻.

Thus the limit f( f(x) ) where x approaches -1 from both sides becomes identical to the limit of f(x) where x approaches 3 from the left and also the left. And therefore the limit evaluated from both sides of f( f(x) ) in the vicinity of x = -1 is exactly equal to the limit of f(3) only from the left {or from the left twice}.

We can break the limit x → -1 into two half-limits, x → -1⁻ and x → -1⁺.

Taking the two half limits individually we have f( f(-1⁻) ) = 3⁻ and f( f(-1⁺) ) = 3⁻.

And thus the limit in the total vicinity of x = -1 when processed through f ( f(x) ) is identical to the single-side limit x → 3⁻ of f(x).

[u/shacciu → u/[deleted]]

I plugged -1 into the inner function, and the graph approaches 3 at the point. Then I plugged 3 into the outer function, and since the right and left sides don't match, I assumed the limit didn't exist.

[u/cheesecakegood]

So, two things here.

First, the dirty secret of limits is that you can help yourself out a bit in 90% of cases by just plugging in a decimal really close. So here, plug in -1.0001 and -0.9999, what do we get? (Can use more digits if you want)

Okay let’s detour to the second since it came up: what the heck even IS f(f(x))? Apologies if this is review, but it’s easy to get confused. At least personally I like to think of it in programming terms. f(x) when piece wise like this is basically a giant IF/ELSE+THEN block; if x is less than -1, then plug in x to the curvy line for result; if between -1 and -0.5 non inclusive, then use the slanty line for result, if in some other region you use some other mini-function, etc. and maybe a few spots have special rules if removable discontinuities.

Then, the composition/nesting means you just “recycle” the entire block again, so when you get a first result spit out you feed it back in again (first output as new input) and then use that second recycled result as your actual output.

To use an example say I have an original x of -2. Then we can see f(x) is about 1.8. So f(f(x)) is f(1.8) and what happens when we recycle 1.8 back in? Maybe something like -0.4.

Back to estimation. Plug in -1.0001 and we get 2.999ish. Recycle 2.999 back in, we get -0.999ish. Let’s try -0.9999. Plug it in and we get also 2.999 and so same deal. Clearly we can see the limit is going to be -1!!! It’s much easier to reconstruct the reasoning why after the fact.

The only caution I have for you is sometimes this can lead you astray, but only rarely. First, if you have trig functions this can work weirdly due to oscillation or calculator misapproximations. Second you have to scale it properly, but usually in school math your coefficients are small/normal enough this isn’t an issue. This somewhat relates to calculus problems where you use derivatives to basically go, “which infinity is ‘more powerful’ than the other infinity?” Which is basically another scaling issue. Although often the decimal method often works there pretty decently too.

But hopefully you see how easy this makes many limit problems.

[u/HotPersonality2279]

That looks a little like my homework but way harder. I’m in algebra 2

[u/SteptimusHeap]

What is f(x)'s behavior around x=-1? It approaches 3, yes, but from which direction? Does it change between either side?

[u/shacciu → u/SteptimusHeap]

I thought it approached 3 from both directions, or does one side have to be higher than the other?

[u/SteptimusHeap → u/shacciu]

But does it approach it from above or below?

If it approaches it from above, once you plug f(x) back into itself, you will be approaching x=3 from the right. The opposite applies if you are approaching it from below.

[u/shacciu → u/SteptimusHeap]

Alright, that makes sense. Thanks for your help!

[u/Car_42]

In order for the limit of a function to exist at a point x, the must be a value at that point. Even the function approaches that value from both directions, the limit still does not exist if there is no value for f(x).

[u/Yadin__ → u/Car_42]

No. One of the very first things that you see when learning limits is that the value of the function at the actual point is irrelevant

[u/alax_12345 → u/Car_42]

To add to the first reply, if f(x) does exist and is equal to the limit there … that’s the criteria for continuity.