We won’t get enough time to spend thinking about all of the possible techniques that we could use to evaluate limits, but in this section we’ll investigate one more.
Ok, trigonometric functions aren’t actually weird. But we want to look at a function that is slightly more complicated than the ones we’ve looked at so far.
This limit is a bit weird, in that we really haven’t looked at trigonometric functions that much. We’re going to start by looking at a different limit in the hopes that we can eventually build towards this one.
Let’s put this limit aside and briefly talk about the sine function. What are some things to remember about this function? What should we know? How does it behave?
In your inequality above, multiply \(\left(\frac{1}{x^2+1}\right)\) onto all three pieces of the inequality. Make sure you’re convinced about the direction or order of the inequality and whether or not it changes with this multiplication.
\begin{equation*}
\underbrace{\frac{\fillinmath{XXXXXXXXXX}}{x^2+1}}_{\text{call this }f(x)} \leq \frac{\sin^2(x)}{x^2+1} \leq \underbrace{\frac{\fillinmath{XXXXXXXXXX}}{x^2+1}}_{\text{call this }h(x)}
\end{equation*}
For your functions \(f(x)\) and \(h(x)\text{,}\) evaluate \(\displaystyle \lim_{x\to\infty} f(x)\) and \(\displaystyle \lim_{x\to\infty} h(x)\text{.}\)
This strategy is a really nice one to use when we know the behavior of some well-behaved “bounding” functions. We can try to off-load the task of summarizing the behavior of a strangely behaved function to these bounding functions, and follow them! As long as they approach each other, than the strangely behaved function has to have the same behavior.
For some functions \(f(x)\text{,}\)\(g(x)\text{,}\) and \(h(x)\) which are all defined and ordered \(f(x)\leq g(x)\leq h(x)\) for \(x\)-values near \(x=a\) (but not necessarily at \(x=a\) itself), and for some real number \(L\text{,}\) if we know that
This theorem can be difficult to use, primarily because building the bounds for a function is difficult. In Activity 1.5.1, we were able to build the boundary functions by simply thinking about the bounds on the \(\sin(x)\text{.}\) This worked well, but we were only able to do this because of our familiarity with this function. With other functions, these bounds are harder to just come up with. This is especially true in that we need the bounds to accomplish multiple things at once:
We need them to be ordered correctly with regard to the function we care about: one above it and one below it.
We need the limits of these functions to be things we can actually evaluate! This is the whole point: we use these (hopefully easier) limits to evaluate the (probably hard) limit that we’re interested in.
Is this limit dependent on the specific version of \(g(x)\) that you sketched? Would this limit be different for someone else’s choice of \(g(x)\) given the same parameters?
For each of the following statements, discuss the possible existence of the limit in question. Explain in detail how we might know whether the limit exists or not, or why it is impossible to tell.
We want to know about \(\displaystyle\lim_{x\to4} f(x) \text{,}\) and we know that \(a(x) \leq f(x) \leq b(x) \) for all \(x \)-values, with \(a(4)=3 \) and \(b(4)=3 \text{.}\)
We want to know about \(\displaystyle\lim_{x\to0} f(x) \text{,}\) and we know that \(\displaystyle\lim_{x\to0} g(x)=5 \) for the function \(g(x)\geq f(x) \) for \(x \)-values around 0.
We want to know about \(\displaystyle\lim_{x\to-1} f(x) \text{,}\) and we know that \(a(x)\leq f(x)\leq b(x) \) for \(x \)-values around \(-1 \text{,}\) with \(\displaystyle\lim_{x\to-1}a(x)=9 \) and \(\displaystyle\lim_{x\to-1}b(x)=9 \text{.}\)
