Limits

A limit describes the process of approaching a point on a graph with a particular value.

There are two types of limits, right-hand limits which approach from the right on the number line and left-hand limits which approach from the left. It is important to note that it is the \( x \) that is moving with respect to \( x_0 \) which remains fixed.

The right-hand limit is written with a plus above the value which \( x \) is approaching.

\[ \lim_{x \to x_0^+} \]

For a right-hand limit \( x > x_0 \) is always true.

The left-hand limit is written with a minus above the value which \( x \) is approaching.

\[ \lim_{x \to x_0^-} \]

For a right-hand limit \( x < x_0 \) is always true.

The right and left hand limits are denoated by the letters \( R \) and \( L \).

For a discontinuous function like

\[ f(x) = \begin{cases} x + 1, & x > 0 \\ -x + 2, & x < 0 \end{cases} \]

For a right-hand limit at \( 0 \) the limit would approach \( 1 \) but the left-hand limit at the same point would approach \( 2 \) because of the discontinuousnature of these equations. The difference of limit from each side is what makes this function discontinuous.

Continuity

A function is continuous if:

\[ \lim_{x \to x_0}f(x) = f(x_0) \]

There are three key requirements to this.

1. That limit for \( L \) and \( R \) (Left and Right limits) exists and that the left and right limits are equal.

2. \( f(x_0) \) is defined.

3. Both the limit and the function of \( x_0 \) are equal the same thing.

Its important to note that even though the function of \( x \) and the limit are equal to the same thing they are not the same thing, they are different processes of arriving at the same value.