Limits in Calculus: The “Close Enough” Answer

limits in calculusThe everyday definition of a limit brings to mind some kind of boundary. When we think of limits we think of a well-defined quantifier with which to work with. We use limits to determine consequences if those limits are crossed and rewards for when we stay within those limits. So why are limits in calculus so troubling for so many students?

Well let’s face it, calculus is an advanced level of mathematics and can be quite intimidating in general.  Studying limits in calculus is just one of many calculus I and calculus II essentials. However, once you can grasp the concept, you’re good to go!

Limits: Almost There but Never There

All of calculus relies on the principle that we can always use approximations of increasing accuracy to find the exact answer, such as approximating a curve by a series of straight lines in differential calculus (the shorter the lines and as the distance between points approaches 0, the closer they are to resembling the curve) or approximating a spherical solid by a series of cubes in integral calculus (as the size of the cubes gets smaller and the number of cubes approaches infinity inside the sphere, the end result becomes closer to the actual area of the sphere).

With the help of modern technology, graphs of functions are often easy to produce. The main focus is between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and long term behavior of a function. In calculus classes, limits are usually the first topic introduced.

In order to understand the workings of much of calculus, we need to understand the concept of a limit. Limits are used in differentiation when finding the approximation for the slope of a line at a particular point, as well as integration when finding the area under a curve. In calculus, limits introduce the component of infinity. We can ask ourselves, what happens to the value of a function as the independent variable gets infinitely close to a particular value?

Show Me Some Examples

Nothing is better than seeing something difficult be put in action.  Take a look at these examples:

  1. f(x) = (x2-1)/(x-1)

Let’s see what happens when we make x=1: f(1) = (12-1)/(1-1) = (1-1)/(1-1) = 0/

Now 0/0 is a problem! We don’t really know the exact value of 0/0, so we need another way of answering this.

Instead of trying to work it out for x=1 let’s try approaching it closer and closer:

x          (x2-1)/(x-1)

0.5         1.50000

0.9         1.90000

0.99       1.99000

0.999     1.99900

0.9999  1.99990

0.99999 1.99999

 

Now we can see that as x gets close to 1, then (x2-1)/(x-1) gets close to 2.

When x = 1 we don’t know the answer, but we can see that it is going to be 2. We want to give the answer “2” but can’t, so instead mathematicians say exactly what is going on by using the special word “limit.” So with this example, we say:

The limit of (x2-1)/(x-1) as x approaches 1 is 2.

It is a special way of saying, “ignoring what happens when you get there, but as you get closer and closer the answer gets closer and closer to 2.”

  1. Consider the limit as x approaches infinity of the function f(x) = 5⁄x

We can find that if we take larger and larger values of x, the value of the fractions becomes smaller and smaller until it gets very close to 0. We say that the limit of 5⁄x as x approaches infinity is 0.

For this function, it is not very obvious what the limit is. We could substitute larger and larger values of x until we see what is happening (try 100, then 1,000, then 10,000, and so on). We could also rearrange the expression and use the fact that the limit as x approaches infinity of 1⁄x is 0 to find the limiting value.

We divide throughout by x to get the expression in a form where we can evaluate it.

Notice that we cannot substitute infinity into the fraction because that does not make mathematical sense (infinity is not a number). The 5⁄x and the 1⁄x go to 0 as x approaches infinity, so those values become 0. We evaluate the limit as -1⁄2.

  1. Next, let’s consider the function f(x)=1⁄x:

What is the behavior of this function as the x value gets bigger? We can see that the graph gets closer to the x axis, which has a height of 0. If we recall in pre-calculus and algebra, this function would have an asymptote at y = 0. We can say that as x approaches infinity, f(x) is approaching 0.

Similarly, we can say that as x approaches negative infinity, it approaches 0 as well.

We can conclude that one over infinity and one over negative infinity both equal 0.

In fact, any number over positive or negative infinity will converge to 0 – unless both the numerator and denominator are positive or negative infinity, then they would converge to 1.

Keep in mind that positive and negative infinity are just ideas. This is why in mathematics notation we use limits to prove as a number gets infinitely big or small, it converges to a number or doesn’t converge at all!

What about when x approaches 0? We can see as it gets closer to the y axis (x=0) from the right it gets really big, and as it approaches the y axis from the left it gets really small.

Since the limit is different from left and from the right, it does not exist. This is why dividing by 0 is undefined – it equals both positive and negative infinity!

Limits are Our Friends

Learning any level of advanced mathematics requires patience and diligence.  Taking the time to discover new ways of learning and anything calculus can be under your belt in no time!  Working your way through the introductory elements of any first year calculus course will set you up for a smoother ride when things get even more abstract! But n0 need to start worrying now.  Limits are our friends and just put in the work.  That’s it.