Help with Limits in Calculus
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 differential and integral 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?
The graph illustrates finding the limit of the dependent variable f(x) as x approaches c. A way to find this is to plug in values that gets close to c from the left and values close to c from the right.
To further illustrate the concept of a limit, consider the sequence of numbers of x:
These values are getting closer and closer to 2 (i.e. they are approaching 2 as their limit). We can can say that no matter what value we consider, 2 is the smallest value that is greater than every output f(x) in the sequence. As we take the differences of these numbers, they will get smaller and smaller. In calculus, the difference between the terms of the sequence and their limit can be made infinitesimally small.
Sometimes, finding the limiting value of an expression means simply substituting a number.
(1) Find the limit as t approaches 10 of the expression
We write this using limit notation as
In this example, we simply substitute and write
There is no complication because M = 3t + 7 is a continuous function, but there are cases where we cannot simply substitute like this.
(2) Find the limit as x approaches 0 of
Notice that we cannot simply substitute 0 because sin(0)⁄0 is undefined, and therefore it is not continuous. There is no algebraic process to find this limit. If we plug in 0 for x, we will get 0⁄0, which is undefined. There is, however, a method using differentiation (see L'Hopital's Rule). We can find the limit without using differentiation by looking at the function's behavior from the left and right of x = 0. We can substitute values that get closer and closer to 0 from the left and the right side to conclude that
A way to check this is to graph it and see that the limit as x gets closer to 0 is 1.
Let's look at a zoomed out view of this image and look at its behavior as x gets infinitely big and infinitely small.
Does this image look familiar? If it does, that's because it is similar to the function of a sound wave, where the x axis is time and the y axis is the amount of decibels (volume). Notice as the the wave trails off in either direction, it approaches 0 but never actually settles. It is interesting to think that every sound wave ever made still exists and is oscillating at an infinitesimally small level!
(3) 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:
(4) Find the limit of this function as x approaches infinity.
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.
(5) Limits may also exist at a point on a graph where the output f(x) is a different value.
We can see that even though the graph is discontinuous as x=2, we know there exists a limit because the graph approaches 2 from the left and the right.
(6) 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 precalculus 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. We can conlude that
This is not possible! 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!
(7) Here is a geometric example of a limit. Let's look at a polygon inscribed in a circle. If we increase the number of sides of the polygon, what can we say about the polygon with respect to the circle?
As the number of sides of the polygon increases, the polygon gets closer and closer
to the becoming the circle. If we generalize the polygon as an
n is the number of sides, we can make some mathematical statements about
- As n gets larger, the n-gon gets closer to being the circle.
- As n approaches infinity, the n-gon approaches the circle.
- The limit of the n-gon as n goes to infinity is the circle.
We can also use differentiation to solve more complex limits, such as indeterminant limits. These are limits where both the numerator and the denominator approach 0 or positive or negative infinity, such as
In the limit on the left, when x approaches 3, the quotient is approaching 0/0. It is not clear what the limit is doing around x = 3. In the limit on the right, as x approaches infinity, the quotient will become ∞⁄∞. Again, it is not quite clear what the limit will be as x approaches infinity. Therefore, these two limits are considered indeterminant.
For solving indeterminant limits, see L'Hopital's Rule.
To recap, limits are concerned with what a function is doing around a given point. We noticed that limits can exist even when the function does not exist at that point. You can find limits by plugging the limiting value into the function (example 1), using tables of values (example 2), and using knowledge of other limits to find the limit of a function at a given point (example 4).
It is important to
- develop an intuitive understanding of the limiting process of a function
- be able to calculate the limit using algebra
- estimate the limit from graphs or tables