Graph each function. Then use your graph to find the...

graph each function. Then use your graph to find the indicated limit, or state that the limit does not exist.
$$f(x)=\left\{\begin{array}{ll}
3 x & \text { if } x<1 \\
x+2 & \text { if } x \geq 1, \lim _{x \rightarrow 1} f(x)
\end{array}\right.$$

Key Concepts

One-Sided Limits

One-sided limits involve approaching a specific point from the left-hand side (values less than the point) or the right-hand side (values greater than the point). This concept is crucial in piecewise functions where the function’s definition changes at the point in question, and comparing these limits determines if an overall limit exists.

Continuity

Continuity at a point means that the function has no breaks or jumps at that point, which requires the left-hand and right-hand limits at the point to exist and be equal, and also equal the value of the function at that point. This concept helps in determining if the limit of a piecewise function at a transition point exists and whether the function is continuous there.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. Understanding these functions involves recognizing which part of the definition applies based on the input value. This concept is essential when evaluating limits because the behavior of the function can change at the points where the definition changes.

Graphing Functions

Graphing functions provides a visual representation of the function’s behavior, allowing one to observe trends, discontinuities, and limits. For piecewise functions, plotting each segment according to its defining condition helps to accurately assess how the function behaves near points of potential transition.

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