Limits of Functions: Definitions and Numerical Approaches

Limits

Definition of the Limit of a Function

The concept of a limit is fundamental in calculus and describes the behavior of a function as its input approaches a particular value. The limit does not necessarily depend on the value of the function at that point, but rather on the values of the function as the input gets arbitrarily close to the point.

• Formal Definition: Suppose the function f(x) is defined for all x near a, except possibly at a itself. If f(x) can be made arbitrarily close to L (as close to L as we like) for all x sufficiently close to a (but not equal to a), then we write:

$\displaystyle \lim_{x \to a} f(x) = L$

• This is read as "the limit of f(x) as x approaches a equals L."

• Note: The value of f(a) (if it exists) does not affect the value of the limit.

Thinking About Limits Numerically

Limits can be explored by evaluating the function at values of x that get closer and closer to a. This approach helps in forming a conjecture about the value of the limit.

• For example, to estimate $\displaystyle \lim_{x \to 0.5} f(x)$, we can compute f(x) for values of x approaching 0.5 from both sides (e.g., 0.4, 0.49, 0.499, 0.5, 0.501, 0.51, 0.6).

• If the values of f(x) get closer to a single number L as x approaches 0.5, we conjecture that the limit is L.

Example Table:

x	f(x)
0.499	0.4999895
0.5	0.5
0.501	0.5000125

As x approaches 0.5 from both sides, f(x) approaches 0.5, suggesting $\displaystyle \lim_{x \to 0.5} f(x) = 0.5$.

One-Sided Limits

Sometimes, it is useful to consider the behavior of a function as x approaches a from only one side.

• Right-Sided Limit: The limit of f(x) as x approaches a from the right (values greater than a) is denoted:

$\displaystyle \lim_{x \to a^+} f(x) = L$

• Left-Sided Limit: The limit of f(x) as x approaches a from the left (values less than a) is denoted:

$\displaystyle \lim_{x \to a^-} f(x) = L$

Relationship Between One-Sided and Two-Sided Limits

The two-sided limit exists if and only if both one-sided limits exist and are equal.

• If $\displaystyle \lim_{x \to a^-} f(x) = L$ and $\displaystyle \lim_{x \to a^+} f(x) = L$, then $\displaystyle \lim_{x \to a} f(x) = L$.

• If the one-sided limits are not equal, the two-sided limit does not exist (DNE).

Example Table:

x (from left)	f(x)	x (from right)	f(x)
0.499	0.4999895	0.501	0.5000125
0.4	0.4	0.6	0.6

If both sides approach the same value, the two-sided limit exists and equals that value.

Summary Table: Types of Limits

Type	Notation	Description
Two-sided limit	$\displaystyle \lim_{x \to a} f(x)$	Approaches a from both sides
Right-sided limit	$\displaystyle \lim_{x \to a^+} f(x)$	Approaches a from the right
Left-sided limit	$\displaystyle \lim_{x \to a^-} f(x)$	Approaches a from the left

Key Points:

• The value of the limit depends on the behavior of the function near the point, not necessarily at the point.

• Numerical and graphical approaches can help estimate limits.

• One-sided limits are essential for understanding discontinuities and piecewise functions.