I hold both a bachelor's and a master's degree in applied mathematics.
The limit of a function f(x) for x to a describes what the function does when you choose x very close to a. Formally, the definition of the limit L of a function is as follows:
For any epsilon greater than 0, there is a delta greater than 0 such that when |x - a| < delta, then |f(x) - L| < epsilon.
This looks complicated but in fact it is not so difficult. What it says is that if we choose x very close to a, namely smaller than delta, we must have that the function value is very close to the limit.
When a is in the domain, this will obviously be just the function value, but the limit might also exist when a is not part of the domain of f.
So, when f(a) exists we have:
limx to a f(x) = f(a)
But the limit can also exist when f(a) is not defined. For example, we can look at the function f(x) = x2/x. This function is not defined for x is 0, since then we would divide by 0. This function does behave exactly the same as f(x) = x at every point except at x = 0, since there it is not defined. Therefore, it is not difficult to see that:
limx to 0 x2/x = 0
Mostly when we talk about limits we mean the two-sided limit. We can however also look at the one sided limit. This means that it is important from what side we "walk over the graph towards x". So we heft the left limit for x to a, which means we start smaller than a and increase x until we reach a. And we have the right limit, which means we start greater than a and decrease x until we reach a. If both the left and right limit are the same we say the (two-sided) limit exists. This does not have to be the case. Look for example at the function f(x) = sqrt(x2)/x.
Then the left limit for x to zero is -1, since x is a negative number. The right limit however is 1, since then x is a positive number. Therefore the left and right limit are not equal, and hence the two-sided limit does not exists.
If a function is continuous in a then both the left and right limit are equal and the limit for x to a is equal to f(a).
The Rule of L'Hopital
A lot of functions will be as the example of the last section. When you fill in a, which was 0 in the example, you get 0/0. This is not defined. These functions do however have a limit. This can be calculated using the rule of L'Hopital. This rule states:
limx to a f(x)/g(x) = limx to a f'(x)/g'(x) if limx to a f(x) = limx to a g(x) = 0 and g'(x) is not equal to 0.
Here f'(x) and g'(x) are the derivatives of these f and g. Our example satisfied all conditions of the l'hopital rule, so we could use it to determine the limit. We have:
f(x) = x2, so f'(x)= 2x
g(x) = x, so g'(x) = 1
Now by the rule of l'hopital we have:
limx to 0 f(x)/g(x) = limx to 0 x2/x = limx to a f'(x)/g'(x) = limx to 0 2x/1 = limx to 0 2x = 0
This limit is 0 because 2x is defined at 0 and is equal to 0 in x=0. So indeed the limit equals zero.
The rule of l'hopital makes use of the derivative. If you are not familiar with the derivative, or if you want to learn more about it I recommend reading my article about the derivative and how to use it.
Limits for x to Infinity
Often we are interested in the behavior of a function when x goes to infinity or minus infinity. This tells us what happens after a very long time, or when we would increase a factor as much as we can. In practice this comes up a lot. Here we need to adjust the definition we gave above. Infinity is not a number so we can not take the difference between x and infinity. The definition is as follows:
For all epsilon greater than zero there is a c greater than zero such that when x is greater than c then |f(x) - L| < epsilon.
So what this means is that if we pick x larger than c then the function value will be very close to the limit value. Such a c must exist for any epsilon, so if someone tells us we must come within 0.000001 from L we can give a c such that f(c) differs less than 0.000001 from L, and so do all function values for x larger than c.
For example the function 1/x has as limit for x to infinity 0 since we can come arbitrarily close to 0 by filling in larger x.
A lot of function go to infinity or minus infinity as x goes to infinity. For example the function f(x) = x is an increasing function and therefore, if we keep filling in larger x, the function will go towards infinity. If the function is something divided by an increasing function in x then it will go to 0.
There are also functions that don't have a limit when x goes to infinity, for example sin(x) and cos(x). These functions will keep oscillating between -1 and 1 and therefore will never be close to one value for all x greater than c.
Properties of Limits of Functions
Some basic properties hold as you would expect for limits. These are:
- limx to a f(x) + g(x) = limx to a f(x) + limx to a g(x)
- limx to a f(x)g(x) = limx to a f(x)*limx to a g(x)
- limx to a f(x)/g(x) = limx to a f(x)/limx to a g(x)
- limx to a f(x)g(x)= limx to a f(x)limx to a g(x)
A special and very important limit is the exponential function. It is used a lot in mathematics and comes up a lot in various applications of for example probability theory. To prove this relation one must use Taylor Series, but that is beyond the scope of this article.
limx to infinity (1+k/x)x = ek
Limits describe the behavior of a function if you look at a region around a certain number. If both one-sided limits exist and are equal, then we say the limit exists. If the function is defined at a, then the limit is just f(a), but the limit might also exist if the function is not defined in a.
When calculating limits, the properties can come up handy, as can the rule of l'hopital.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.