Limits are one of the foundational components of Calculus. A limit is the value that a function approaches rather than what it is equal to.
Limits can be expressed algebraically as:
You read this as "the limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(L\)".
Let's see how limits are different from just evaluating the function. For example, let's try finding the output of \(\cfrac{x^2-1}{x-1}\) at \(x = 1\):
As you can see, when we try evaluating the function at this value, we get an error. This means that the function is not defined at \(x = 1\). In this case, there is a hole in the function at this point. We can see the hole when we plot the function:
However, if we consider the limit of the function as \(x\) approaches \(1\), we would say that the function apporaches a value of \(2\) even though the function does not exist at \(x = 1\). The limit says that the function approaches \(2\) from both sides when you are near \(x = 1\). We can write this as:
In this example, the white circle at \(x = 1\) indicates that the function can approach this point but never actually touches it.
The \(x\)-value can approach \(a\) on both the left and right sides, referred to as Left-Handed Limits and Right-Handed Limits respectively. These limits are denoted with negative and positive superscripts on the value \(a\) respectively. If the Left and Right-Hand Limits both approach the same value, then the function contains a Two-Sided Limit.
A limit can exist if the following 3 conditions are satisfied:
A limit will not exist under the following conditions:
1. There is a jump in the graph
We can determine the following limits from this graph:
Since both sides don't approach the same value, the limit doesn't exist.
2. There is a one-sided graph
We can determine the respective Right and Left-Handed Limits as such:
Since the function only moves in one direction, a Left-Handed Limit does not exist. As a result, a limit doesn't exist for the function.
3. The graph approaches ∞ due to a Vertical Asymptote
We can determine the respective Right and Left-Handed Limits as such:
Since both sides approach infinity, we can determine that a limit doesn't exit.
Find the limits for \(a = 1\) and \(a = 5\) for the following graph. Determine if either limit is a 2-Sided Limit:
We can determine the following limits for \(a = 1\):
We can determine the following limits for \(a = 5\):
We can determine that \(\lim\limits_{x\to5} f(x) = 4\) is a Two-Sided limit since both the Left and Right Sides approach the same value.
\(f(x) = \cfrac{1}{x+2}\) at \(x = -3\)
\(g(x) = \sqrt{4-x}\) at \(x=4\)