Limits and Continuity

This section covers:


Note that we discuss finding limits using L’Hopital’s Rule here.

We need to understand how limits work, since the first part of Differential Calculus (calculus having to do with rates at which quantities change)   uses them.   I like to think of a limit as what the y part of a graph or function approaches as x gets closer and closer to a number, either from the left hand side (which means that x part is increasing), or from the right hand side (which means the x part is decreasing).

We can write a limit where x gets closer and closer to 0 as  \(\underset{{x\to 0}}{\mathop{{\lim }}}\,f\left( x \right)=L\).   To describe this, we say the “limit of  \(f\left( x \right)\)  as x approaches 0 is L”.   Now the beauty of limits is that x can get closer and closer to a number, but not actually ever get there (think asymptote), or it could get there, and it would still be a limit!

The reason we have limits in Differential Calculus is because sometimes we need to know what happens to a function when the x gets closer and closer to a number (but doesn’t actually get there); we will use this concept in getting the approximation of a slope (“rate”) of a curve at that point.    Sometimes, the x value does get there (like when we’re taking the slope of a straight line), but sometimes it doesn’t (like when we’re taking the slope of a curved function).

As an example, when you first learn how to handle limits, it might be the case that the x value is getting closer and closer to a number that makes the denominator of the y value 0; this can’t happen, or the fraction will “blow up”.

Here’s what a limit might look like graphically; note that when we factor this function (difference of cubes), we find we have a removable discontinuity, or hole:

Limits Graphically

Finding Limits Algebraically

We will learn different techniques for finding simple limits; here are some (non-trig) problems:

Finding Limits Algebraically

Continuity and One Side Limits

Sometimes, the limit of a function at a particular point and the actual value of that function at the point can be two different things.  Notice in cases like these, we can easily define a piecewise function to model this situation.

The limit from the right,  or  \(\underset{{x\to {{c}^{+}}}}{\mathop{{\lim }}}\,f\left( x \right)=L\)  means that x approaches c from the right side, or with values greater than c, and the limit from the left,  or  \(\underset{{x\to {{c}^{-}}}}{\mathop{{\lim }}}\,f\left( x \right)=L\)   means that x approaches c from the left side, or with values less than c.   Do you see how if the limit from the right and the limit from the left are the same, then we can get a “regular” limit (meaning both sides converge to the same y value?)

Existence of a Limit and Definition of Continuity

Do you also see that if the limit from the right equals the limit from the left, and this equals the actual point for  \(f\left( x \right)\)   (the y for that x), then we have a continuous function (one that we can draw without picking up our pencil)?   This leads to the definition of the existence of a limit, the formal definition of continuity:

Definition of Limits and Continuity

Here are some examples; remember that the actual limits are the y values, not the x.  The first example shows that some limits do not exist (DNE), based on the definition above.  The second example actually gives you the equation for the piecewise function that illustrates limits.  Notice that both functions are discontinuous.

Graphs Showing Limits

Continuity of Functions

We learned in the Graphing Rational Functions, including Asymptotes section how to find removable discontinuities (holes) and asymptotes of functions (basically anywhere where we’d get a 0 in the denominator of the function);  now we know that these functions are discontinuous at these points.

Let’s review how we get vertical asymptotes for a rational function:

holes-and-vertical-asymptotes

And we’ll also have to remember the trig function asymptotes:

Trig Function Asymptotes

In Calculus, you may be asked to find the x-values at which a function might be discontinuous, and also determine whether or not a discontinuity is removable or non-removable:

Continuity

Property of Limits

Limits have properties that are pretty straightforward; basically these just mean that you can add, subtract, multiply, and divide limits (and multiply them by a number, or scalar) with the limit  on the “inside” or “outside”.  (Think about “picking apart” limits into smaller pieces.)  And remember again that the limits refer to the “y” variable.

The properties are:

  1. \(\underset{{x\to c}}{\mathop{{\lim }}}\,\left[ {b\cdot f\left( x \right)} \right]=b\cdot \underset{{x\to c}}{\mathop{{\lim }}}\,f\left( x \right)\)      (Scalar Multiple)
  2. \(\underset{{x\to c}}{\mathop{{\lim }}}\,\left[ {f\left( x \right)\pm g\left( x \right)} \right]=\underset{{x\to c}}{\mathop{{\lim }}}\,f\left( x \right)\pm \underset{{x\to c}}{\mathop{{\lim }}}\,g\left( x \right)\)       (Sum or Difference)
  3. \(\underset{{x\to c}}{\mathop{{\lim }}}\,\left[ {f\left( x \right)\cdot g\left( x \right)} \right]=\underset{{x\to c}}{\mathop{{\lim }}}\,f\left( x \right)\cdot \underset{{x\to c}}{\mathop{{\lim }}}\,g\left( x \right)\)       (Product)
  4. \(\underset{{x\to c}}{\mathop{{\lim }}}\,\frac{{f\left( x \right)}}{{g\left( x \right)}}=\frac{{\underset{{x\to c}}{\mathop{{\lim }}}\,f\left( x \right)}}{{\underset{{x\to c}}{\mathop{{\lim }}}\,g\left( x \right)}}\)       (Quotient)
  5. \(\underset{{x\to c}}{\mathop{{\lim }}}\,\left[ {f{{{\left( x \right)}}^{n}}} \right]={{\left[ {\underset{{x\to c}}{\mathop{{\lim }}}\,f\left( x \right)} \right]}^{n}}\)       (Power)
  6. \(\underset{{x\to c}}{\mathop{{\lim }}}\,f\left( {g\left( x \right)} \right)=f\left( {\underset{{x\to c}}{\mathop{{\lim }}}\,\,g\left( x \right)} \right)\)        (Composite Functions)


Here is an example of how the sum property of limits works:  \(\underset{{x\to 1}}{\mathop{{\lim }}}\,\,\,\left( {5{{x}^{2}}+2x-1} \right)=\underset{{x\to 1}}{\mathop{{\lim }}}\,\,5{{x}^{2}}+\underset{{x\to 1}}{\mathop{{\lim }}}\,\,2x-\underset{{x\to 1}}{\mathop{{\lim }}}\,\,1=5+2-1=6\).

Limits with Sine and Cosine

There are a couple of special trigonometric limits that you’ll need to know, and to use these, you may have to do some algebraic tricks.   These are the two limits to learn:

Limits with Sine and Cosine

Note that for the first limit (with sin), the reciprocal is also true, since  \(\frac{1}{1}=1\).

Here are the types of problems you might see.  Note again that you can check these in your calculator by putting in numbers really close to the x values in your calculator (such as x = .00001 for x approaches 0).

Limits of Sin Functions

Intermediate Value Theorem (IVT)

The intermediate value theorem (IVT) seems very complicated and is a bit theoretical, but if we think about what it really says, it’s not that difficult and pretty obvious.

What the intermediate value theorem says is that if you are at a certain x (where you have a y value)and you go to another value x to the right (where you have another y value), and the path that you go is on a continuous function, then you have to have hit (cross over) all the y values in between.

So that still sounds confusing, so let’s think of an example.  Let’s say a baby boy weighs 7 pounds at birth, and then 20 pounds when he is 1 year old (12 months).   At some point, he must have weighed say 15 pounds, or actually any number of pounds between 7 and 20 pounds.   So the baby’s age is the x value, and the baby’s weight is the y value in this case, with the interval being between 0 and 12 months, inclusive.   This makes since a human weight is continuous, and it doesn’t jump up or down instantaneously.

The other way to think of IVT is that we have 2 points on a continuous curve and there is a horizontal line between these two points, then the curve must cross this horizontal line to get from one point to the other point.

Here is the formal definition (and picture) of the Intermediate Value Theorem:

IVT

Here are some types of problems that you might see with the Intermediate Value Theorem:

Intermediate Value Theorem Examples Infinite Limits

An infinite limit is just a limit in which the y either increases or decreases without bound (goes up forever or down forever) as x gets closer and closer to a value.    We typically think of these types of limits when we deal with vertical asymptotes (VA’s), so we can use what we know about VA’s to work with them.

We learned earlier how to get the vertical asymptotes of functions, including trig functions.

When a function gets closer and closer to a VA, the limit will either be  \(-\infty \)  or \(-\infty \).   To determine which one it is, we can put in numbers (for x) really close to the VA on either side, or by using a graphing calculator.

Let’s do some problems where we need to find the one-sided limit (if it exists).  Some of these may involve remembering rational parent functions.  You can also try these on your graphing calculator to get the answers.

Infinite Limits

Limits at Infinity

Limits at Infinity exist when the x values (not the y) go to  \(\infty\)  or  \(-\infty \),  so when we have rational functions.  This is because when we have rational functions,  we’re usually dealing with a horizontal asymptote (HA) (also called an end behavior asymptote, or EBA).   The y values can get closer and closer to a number, but never actually reach that number in the case of an EBA.    Let’s review how to get horizontal or end behavior asymptotes:

Calculus Review of Horizontal Asymptotes

The easiest way to get limits at infinity with rational functions is to find the horizontal asymptotes (end behavior asymptotes) in that direction.  We can  also use a trick where we can divide every term in the numerator and demoninator by the variable with the highest degree (highest exponent value).  This is because of the following Limits of Infinity Theorems:

Limits at Infinity

Basically all this says is that if the bottom (numerator) of a fraction gets bigger and bigger (towards   \(\infty \)  or  \(-\infty \)), the whole fraction will get smaller and smaller and eventually go to 0.

When we do limit problems where there are x’s on the top and bottom, when we try to plug in  \(\infty \)  or  \(-\infty \),  we’ll typically get what we call indeterminate form – something like  \(\frac{\infty }{\infty }\).   So we’ll have to use the tricks of finding the horizontal asymptote, or dividing all the terms by the variable with the highest degree.    Sometimes if we have roots in the function, we can multiply by the conjugate of the numerator or denominator and try to go from there.

Let’s do some problems.    Note that you can check these by trying to put in a large number (or very small number for  \(-\infty \))  for x in your graphing calculator.

Limits at Infinity Problems

Learn these rules, and practice, practice, practice!


Use the MathType keyboard to enter a Limit problem, and then click on Submit (the arrow to the right of the problem) to solve the problem. You can also click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Definition of the Derivative  – you are ready!