Numerical and graphical approaches are used to introduce the concept of limits using examples.
In what follows, \( C \) represents the constant of integration where applicable.
Numerical Approach to Limits
Example 1
Let \( f(x) = 2x + 2 \) and compute \( f(x) \) as \( x \) takes values closer to 1. We first consider values of \( x \) approaching 1 from the left (\( x < 1 \)).
We now consider \( x \) approaching 1 from the right (\( x > 1 \)).
In both cases as \( x \) approaches 1, \( f(x) \) approaches 4. Intuitively, we say that \( \lim_{x \to 1} f(x) = 4 \).
NOTE: We are talking about the values that \( f(x) \) takes when \( x \) gets closer to \( 1 \) and not \( f(1) \). In fact, we may talk about the limit of \( f(x) \) as \( x \) approaches \( a \) even when \( f(a) \) is undefined.
Example 2
Let \( g(x) = \dfrac{\sin x}{x} \) and compute \( g(x) \) as \( x \) takes values closer to 0. We consider values of \( x \) approaching 0 from the left (\( x < 0 \)) and values of \( x \) approaching 0 from the right (\( x > 0 \)).
Here we say that \( \lim_{x \to 0} g(x) = 1 \). Note that \( g(0) = \dfrac{\sin 0}{0} = \dfrac{0}{0} \) is undefined at \( x = 0 \).
Graphical Approach to Limits
Example 3
The graph below shows that as \( x \) approaches 1 from the left, \( y = f(x) \) approaches 2 and this can be written as:
\[ \lim_{x \to 1^-} f(x) = 2 \]As \( x \) approaches 1 from the right, \( y = f(x) \) approaches 4 and this can be written as:
\[ \lim_{x \to 1^+} f(x) = 4 \]Note that the left and right hand limits and \( f(1) = 3 \) are all different.
Example 4
This graph shows that:
\[ \lim_{x \to 1^-} f(x) = 2 \]As \( x \) approaches 1 from the right, \( y = f(x) \) approaches 4 and this can be written as:
\[ \lim_{x \to 1^+} f(x) = 4 \]Note that the left-hand limit \( \lim_{x \to 1^-} f(x) = 2 \) and \( f(1) = 2 \) are equal.
Example 5
This graph shows that:
\[ \lim_{x \to 0^-} f(x) = 1 \]and
\[ \lim_{x \to 0^+} f(x) = 1 \]Note that the left and right hand limits are equal and we can write:
\[ \lim_{x \to 0} f(x) = 1 \]In this example, the limit when \( x \) approaches 0 is equal to \( f(0) = 1 \).
Example 6
This graph shows that as \( x \) approaches -2 from the left, \( f(x) \) gets smaller and smaller without bound and there is no limit. We write:
\[ \lim_{x \to -2^-} f(x) = -\infty \]As \( x \) approaches -2 from the right, \( f(x) \) gets larger and larger without bound and there is no limit. We write:
\[ \lim_{x \to -2^+} f(x) = +\infty \]Note that \( -\infty \) and \( +\infty \) are symbols and not numbers. These are symbols used to indicate that the limit does not exist.
Example 7
The graph below shows a periodic function whose range is given by the interval \([-1, 1]\). If \( x \) is allowed to increase without bound, \( f(x) \) takes values within \([-1, 1]\) and has no limit. This can be written:
\[ \lim_{x \to +\infty} f(x) = \text{does not exist} \]If \( x \) is allowed to decrease without bound, \( f(x) \) takes values within \([-1, 1]\) and has no limit again. This can be written:
\[ \lim_{x \to -\infty} f(x) = \text{does not exist} \]
Example 8
If \( x \) is allowed to increase without bound, \( f(x) \) in the graph below approaches 2. This can be written:
\[ \lim_{x \to +\infty} f(x) = 2 \]If \( x \) is allowed to decrease without bound, \( f(x) \) approaches 2. This can be written:
\[ \lim_{x \to -\infty} f(x) = 2 \]
