Limits and Continuity

Limits

Evaluating Limits

Consider the following three functions: \[f(x) = x + 1, \; g(x) = {x^2 + x\over x}, \; h(x) = {x^2 + 2x + 1\over x + 1}\] Sketch a graph of all three functions (or throw them into a graphing program). What do you notice? What happens at \(x = 0\) or \(x = -1\) for all three functions? \(g(x)\) has a hole at \(x = 0\). But what about \(g(0.000000001)\) or \(g(-{1 \over 10^{100}})\)? Those have values even though they are so very close to the hole at \(x=0\).

Since we are studying change, we are more concerned with how these functions behave as we approach certain points rather than specific values. So while it is still essential to acknowledge the hole at \(h(-1)\), we are also concerned with what happens as the inputs of the function trend towards those hole values as well. This brings us to limits! \[\lim_{x\to 0} f(x) = \lim_{x\to 0} g(x) = \lim_{x\to 0} h(x)\]

Informally, a limit describes what output a function leads to as it approaches a certain input. For the examples above, it shows what happens as the \(x\) term approaches \(0\). Slightly less informally, it describes what value it approaches from both directions, as it increases towards \(0\) from the left (\(\lim_{x\to 0^-}\)) and decreases towards it from the right (\(\lim_{x\to 0^+}\)). Note that the sign shows the direction to approach from, not the direction of travel.

For some cases limits can be solved visually with the graph or algebraically. For the three examples above, a plot clearly shows them all approaching \(y = 1\) from both directions. You could reduce \(g(x)\) and \(h(x)\) to get them to be identical to \(f(x)\) (though this would be bad practice in other contexts since it is rarely a good idea to eliminate a variable term). When solving limits that approach a set value these methods are usually fairly effective. The visual of the graph can show that the two sides approach some value (or else the limit does not exist) and if the input value results in a hole or asymptote, there could be different methods to eliminate that like factoring or properties of whatever transcendental functions are involved in the problem. Not to mention all of these methods are only necessary when plugging in the \(x\) term directly doesn’t just lead to a solution!

A formal definition of a limit exists, though it is beyond the scope of most Calculus courses and the informal logic works just as well. Suppose we want to show the following: \[\lim_{x\to c}f(x)=L\] Then for all \(x\) values, except \(c\), there are positive quantities \(\epsilon\) and \(\delta\) where: \[|f(x) - L|\lt\epsilon\ \; \text{if} \; 0\lt|x-c|\lt\delta\]

The formal definition supposes two quantities: epsilon (\(\epsilon\)), which is compared to the distance between the function’s output and the solution to the limit (the \(y\) values), and delta (\(\delta\)), which is compared the distance between the function’s input and the target value of the limit (the \(x\) values). Note that this definition supposes we have a solution we are verifying; this is not an algorithm to solve the problem.

Infinite Limits

Let’s consider two more functions: \[a(x) = {1\over x}, \; b(x) = {1\over x^2}\] Sketch these two functions and explore their behavior as they approach \(0\). Notice that for \(a(x)\): \[\lim_{x\to 0^-} a(x) \neq \lim_{x\to 0^+} a(x)\] This means that \(\lim_{x\to 0} a(x)\) does not exist (DNE) since the one-sided limits do not match! But now look at b(x): \[\lim_{x\to 0^-} b(x) = \lim_{x\to 0^+} b(x) = \infty = \lim_{x\to 0} b(x)\]

This should be fairly intuitive looking at the graph for both functions, but what if you want to solve limit problems without a visual? You could explore different \(x\) inputs as they approach the target from the direction of interest. For example, for \(\lim_{x\to 0^-}a(x)\) you could try \(a(-1)\), \(a(-0.1)\), \(a(-.001)\), and so on until a pattern jumps out at you!

Limits to Infinity

Another important use of limits is to explore what happens when functions move towards infinity. Take for example: \[\lim_{x\to \infty}a(x)\;\text{and}\lim_{x\to \infty}b(x)\] Visually, both of these functions will tend towards \(0\) and plugging in increasingly large values seconds that. Note that limits to infinity are not concerned with the rate at which a function heads towards some value (except in the case of two functions divided by each other, which is the next topic). For example, consider how: \[\lim_{x\to \infty}\ln(x) = \infty\] This graph might be increasing at a slowing rate, but it is increasing nonetheless!

Dealing with infinite limits of quotients can be a fun logic puzzle! Consider: \[\lim_{x\to \infty}{x^7 + 3x^4 - 2x + 1\over 5x^4 - 2x^3 -2}\] Our toolbox so far doesn’t present a lot of options. There are no efficient ways to factor this and it isn’t even clear if there are any holes or asymptotes to consider at first glance. But since we are going to infinity, we don’t need to really worry about all the small things! Instead, let’s focus only on the leading terms in the numerator and denominator (assuming the function is written in order): \(x^7\) and \(5x^4\). As we tend towards infinity, those will be the only significant terms for a couple of reasons. First is that if we were to “plug in” \(\infty\) to the function, those have the highest powers and even though it's nonsensical to say multiplying by more infinities makes something larger, it does have a logic to it. More rigorously, we could divide both the top and bottom of the fraction by \(x^7\) which will leave most terms with negative exponents on the \(x\) and anything divided by infinity will tend away towards \(0\). This would leave us with a leading \(x^3\) term, which as a positive, odd function will be increasing towards infinity which leads to a solution of \(\infty\).

The Squeeze Theorem

Periodic functions like \(\sin()\) can be a problem for the limit strategies covered above, especially when paired with indeterminate situations like division by \(x\). Though cases like \(\lim_{x\to \infty}\sin(x)\) intuitively do not exist, some problems will need a clever approach: \[\lim_{x\to 0}x^{2}\sin({1\over x})\] Before spoiling the strategy (like the section title already has), let’s try to approach from scratch. With a \(\sin()\) function, the \(\sin()\) term will account for the periodic motion, but the term(s) outside of that will shape its amplitude. So this function is really just \(\pm x^2\) serving as the bounds for a function oscillating due to the \(\sin({1\over x})\) term. Plotting that makes a delightful phenomena clear at \(x = 0\): this function is also tending towards \(0\).

This is the guiding logic behind The Squeeze Theorem! If there are two solvable functions that both trap and ensnare a function at the target value, then you can discover that original function’s limit! Figuring out the two “bread” functions for the limit sandwich can be a bit tricky (see \(\lim_{x\to 0}{\sin(x)\over x} = 1\)) but in most settings the process leading to the limit problem can highlight potential contenders.

Continuity

Defining Continuity

When we collect data, we only measure a sample of points like a runner’s position at different times or locations. We then infer the spaces between those points with a significant assumption: continuity. This isn’t an absurd assumption to make - the runner is probably not teleporting across the track - but it does carry a lot of weight for what we can do mathematically. We need a way to define continuity without having to rely on a mechanical explanation for whatever data collection process we are using. A function is considered continuous at a point, say \(c\), if: \[\lim_{x\to c}f(x) = c\] This definition can be expanded to endpoints or domains, but that equality has to hold throughout. If not, there is a discontinuity where \(\lim_{x\to c}f(x) \ne c\) and the function is called discontinuous in that domain.

Removing Discontinuities

Let's revisit the functions from the start of the unit: \[f(x) = x + 1, \; g(x) = {x^2 + x\over x}, \; h(x) = {x^2 + 2x + 1\over x + 1}\] \(f(x)\) is continuous for any \(x\) but that is not the case for \(g(x)\) or \(h(x)\). What if we wanted to make them continuous? First we have to identify the point(s) that cause the discontinuity. For \(g(x)\), \(\lim_{x\to 0}\ne g(0)\) and for \(h(x)\), \(\lim_{x\to -1}\ne h(-1)\). To fix those holes, we would have to redefine \(g(0)\) and \(h(-1)\) to be equal to the limit at those points - which is something you already know! So how do we put that together? With piecewise functions: \[g(x) = \begin{cases} 1 & x = 0\\ {x^2 + x\over x} & x\ne 0\end{cases}\]

The Intermediate Value Theorem

If a function is continuous for the domain we are interested in, we can make some useful assumptions about it. Let’s say you set out a bucket to measure how much rain falls overnight in your area. You measure it right before going to bed at 2.32 cm and wake up to see that the water level is now 3.53 cm. You can confidently state that the bucket had every measurement possible between those two values overnight! Could a raccoon have come by to sip some, leaving it a bit below 2.32 cm? Maybe! Could it have gone past 3.53 cm and some evaporated before you saw it? Perhaps! But was there a time where it had 2.75 cm of water? Definitely!

That’s the guarantee provided by The Intermediate Value Theorem! If a function is continuous between two points, say the domain \([a, b]\), then we can confidently say that every output value between \(f(a)\) and \(f(b)\) exists! This doesn’t mean other values can’t also be present in that domain, just that the range between those results is certain. This lets us confirm lots of concepts in science with a higher level of confidence. For example if an object goes from a negative velocity to a positive one, the IVT shows that there is a moment, even if it is only for an instant, where that object must be at rest with a velocity of zero.

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