91Ó°ÊÓ

Limits are a fundamental concept in calculus and help us understand how a function behaves as it approaches a specific value. This is especially crucial when dealing with piecewise functions, where the function's formula changes based on the input value. In our exercise, we encounter the piecewise function \( g(t) \) with different expressions depending on the range of \( t \). The limit \( \lim_{t \to a} g(t) \) is found by considering the values \( g(t) \) approaches as \( t \) gets close to \( a \) from both the left (\( t^{-} \)) and the right (\( t^{+} \)).

• For \( \lim_{t \to 0} g(t) \), the left-hand limit \( g(t) = t - 2 \) gives \( -2 \), while the right-hand limit \( g(t) = t^2 \) gives \( 0 \). These limits are not equal, so \( \lim_{t \to 0} g(t) \) does not exist.
• For \( \lim_{t \to 1} g(t) \), we focus on the interval \( 0 \leq t \leq 2 \) where \( g(t) = t^2 \). The limit thus equals \( 1^2 = 1 \).
• For \( \lim_{t \to 2} g(t) \), the left limit using \( g(t) = t^2 \) and the right limit using \( g(t) = 2t \) both approach 4. So, \( \lim_{t \to 2} g(t) = 4 \).

Understanding limits provides insight into the function's behavior and helps determine continuity and points of interest like jumps or asymptotes.

Continuity at a point means that a function does not have any gaps, jumps, or breaks at that point. It's like drawing the graph without lifting your pencil. For a function \( g(t) \) to be continuous at a point \( t = a \), three conditions must be satisfied:

• The limit \( \lim_{t \to a} g(t) \) must exist.
• The function value \( g(a) \) must be defined.
• The limit must equal the function value, \( g(a) = \lim_{t \to a} g(t) \).

In our piecewise function scenario:- At \( t = 0 \), \( g(t) \) is not continuous because the limit does not exist. There's a jump between \( -2 \) and \( 0 \).- At \( t = 1 \), although the limit \( \lim_{t \to 1} g(t) = 1 \) and \( g(1) = 1 \), the function is continuous because both values are the same and no jump exists.- At \( t = 2 \), \( \lim_{t \to 2} g(t) = 4 \) matches \( g(2) = 4 \), indicating the function is continuous there since both conditions are met.

Understanding continuity helps identify where a function behaves predictably without jumps or breaks.

Solving calculus problems like the one involving piecewise functions requires systematic approaches. Let's consider a few strategies to ensure accuracy and understanding:

• **Analyze the function:** Break down the piecewise function into its respective intervals and formulas. Know which expression to use for each range of \( t \).
• **Determine limits:** Use left-hand and right-hand approaches to calculate limits at points of interest. When left and right limits are unequal, note that the overall limit doesn't exist.
• **Check for continuity:** Apply the three conditions for continuity at key points to see if the function graphs smoothly without interruption.

Using these strategies, dealing with each segment of \( g(t) \) separately ensures that no steps are missed. This improves your skill in solving calculus problems systematically and logically by understanding the behavior of piecewise functions across different intervals.