Limit of a Function#
Definition of a Limit#
Definition
The limit of \(f(x)\) as \(x\) approaches \(a\) is equal to the finite number \(L\), denoted by
if the value of \(f(x)\) can be made as close to \(L\) as we want by taking x values sufficiently close to, but not equal to, \(a\).
Keep in mind that the value of a limit does not depend on the value of the function at \(x=a\). In fact, it is possible that \(\lim\limits_{x\to a}f(x) = L\) even though
\(f(a)\) does not exist, or
\(f(a)\) exists, but is not equal to \(L\).
Exercise
Below, you see a table in which we evaluate a function \(f(x)\) for values of \(x\) that are closer and closer to \(1\) (but not equal to \(1\)). What do the values \(f(x)\) suggest \(\lim\limits_{x\to 1} f(x)\) would be?
\(x\) |
\(f(x)\) |
|---|---|
1.100000 |
4.63000000 |
1.010000 |
4.06030000 |
1.001000 |
4.00600300 |
1.000100 |
4.00060003 |
Show solution
The values \(f(x)\) get closer to \(4\) as \(x\) gets closer to \(1\). This suggests that \(\lim\limits_{x \to 1} f(x) = 4\).
One-Sided Limits#
Definition
The limit of \(f(x)\) as \(x\) approaches \(a\) from the right is equal to the finite number \(L\), denoted by
if the values of \(f(x)\) can be made as close to \(L\) as we want by taking \(x\) sufficiently close to (but not equal to) \(a\) and to the right of \(a\) (i.e., \(x>a\)).
Similarly, the limit of \(f(x)\) as \(x\) approaches \(a\) from the left is equal to the finite number \(M\), denoted by
if the values of \(f(x)\) can be made as close to \(M\) as we want by taking \(x\) sufficiently close to (but not equal to) \(a\) and to the left of \(a\) (i.e., \(x<a\)).
Exercise
Below, you see a table in which we evaluate a function \(f(x)\) for values of \(x\) that are closer and closer to \(1\) (but not equal to \(1\)). The first four values are for inputs \(x < 1\), the next four are for inputs \(x> 1\).
\(x\) |
\(f(x)\) |
|---|---|
0.900000 |
1.810000 |
0.990000 |
1.980100 |
0.999000 |
1.998001 |
0.999900 |
1.999800 |
1.000100 |
1.000600 |
1.001000 |
1.006003 |
1.010000 |
1.060300 |
1.100000 |
1.630000 |
What do the values \(f(x)\) suggest \(\lim\limits_{x\to 1^+} f(x)\) and \(\lim\limits_{x\to 1^-} f(x)\) would be?
Show solution
If approaching from the left (\(x< 1\)), the values \(f(x)\) get closer to \(2\) as \(x\) gets closer to \(1\). This suggests that \(\lim\limits_{x \to 1^-} f(x) = 2\).
If approaching from the right (\(x> 1\)), the values \(f(x)\) get closer to \(1\) as \(x\) gets closer to \(1\). This suggests that \(\lim\limits_{x \to 1^+} f(x) = 1\).
Limit vs. One-Sided Limits#
Fact
If both one-sided limits exist and are equal to the same value, \(L\), then we can say that the limit exists, and is also equal to \(L\). In other words,
means the same thing as