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
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

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 00,
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)
= 5x

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 5x 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 1x
is 0 to find the limiting value.

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

Notice that we cannot substitute infinity into the fraction because that does not
make mathematical sense (infinity is not a number). The 5x
and the 1x go to 0 as x approaches infinity,
so those values become 0. We evaluate the limit as -12.

(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)=1x:

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

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

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-gon, where
n is the number of sides, we can make some mathematical statements about
the polygon:

  • 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
. 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

Limits Summary

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

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