Estimating Limits from Graphs and Understanding One-Sided Limits

The graph of a function provides a visual way to estimate limits. From the graph, you can identify how the function behaves as \( x \) approaches a particular value from the left or from the right.

This leads to the concepts of Left-Hand Limit (LHL) and Right-Hand Limit (RHL).

Definitions:

Left-Hand Limit (LHL):

\( \lim_{x \to a^-} f(x) \) is the value the function approaches as \( x \) approaches \( a \) from the left (smaller values of \( x \)).

Right-Hand Limit (RHL):

\( \lim_{x \to a^+} f(x) \) is the value the function approaches as \( x \) approaches \( a \) from the right (larger values of \( x \)).

The two-sided limit \( \lim_{x \to a} f(x) \) exists if and only if the LHL and RHL both exist and are equal.

Limits at Infinity (One-Sided):

• \( \lim_{x \to \infty} f(x) \): Describes the behavior of \( f(x) \) as \( x \) grows very large (to the right on the graph).
• \( \lim_{x \to -\infty} f(x) \): Describes the behavior of \( f(x) \) as \( x \) becomes very negative (to the left on the graph).
• These limits are often used to identify horizontal asymptotes.

Key Observations from Graphs:

• If the graph approaches the same height from both sides near \( x = a \), the limit exists.
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• If the left and right behaviors differ, the two-sided limit does not exist, but one-sided limits may exist individually.
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• If the graph goes to infinity or oscillates near \( x = a \), the limit does not exist.
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Example:

The graph of the piecewise function is given as:

\(f(x) = \begin{cases} 3x, & x < 0 \\ 12 – 2x, & x \ge 0 \end{cases} \)

Using the graph, find:

1. \( \lim_{x \to 0^-} f(x) \)
2. \( \lim_{x \to 0^+} f(x) \)
3. Does \( \lim_{x \to 0} f(x) \) exist?