For the function f whose graph is given, state the value of...

For the function f whose graph is given, state the value of the given quantity if it exists. If it does not exist, explain why. $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 3} f(x)} & {\text { (b) } \lim _{x \rightarrow 1} f(x)} & {\text { (c) } \lim _{x \rightarrow-3} f(x)} \\ {\text { (d) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (e) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (f) } \lim _{x \rightarrow 2} f(x)}\end{array} $$

For the function f whose graph is given, state the value of the given quantity if it exists. If it does not exist, explain why.
$$
\begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 3} f(x)} & {\text { (b) } \lim _{x \rightarrow 1} f(x)} & {\text { (c) } \lim _{x \rightarrow-3} f(x)} \\ {\text { (d) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (e) } \lim _{x \rightarrow 2^{-}} f(x)} & {\text { (f) } \lim _{x \rightarrow 2} f(x)}\end{array}
$$

Key Concepts

Graph Interpretation in Calculus

Interpreting a function’s graph is a practical method to visually assess limits. By observing the behavior of the graph as it approaches a given x-value, one can determine one-sided limits and the overall limit. This technique is essential for evaluating the behavior of functions at points of potential discontinuity or while dealing with piecewise-defined functions.

Continuity and Discontinuities

Continuity requires that a function’s limit as the input approaches a point exists and is equal to the function's value at that point. When the left-hand and right-hand limits at a point do not match, or the limit does not equal the function's value, a discontinuity exists. Jump discontinuities, for example, occur when the one-sided limits exist but are not equal, leading to the overall limit failing to exist.

Limit of a Function

The concept of a limit involves determining the value that a function approaches as the input nears a particular point. It focuses on the behavior of the function close to that point rather than the function's actual value at the point. This is fundamental in calculus because it helps in understanding continuity and in defining derivatives.

One-Sided Limits

One-sided limits refer to the values that a function approaches as the input approaches a specified point from one side only – either from the left (denoted as x?c?) or from the right (denoted as x?c?). These are particularly useful when the behavior of the function is not symmetrical around the point or when there's a discontinuity, ensuring a precise evaluation of the limit from one direction.

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