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For the function g whose graph is given, state the value of
each quantity, if it exists. If it does not exist, explain why lim t 2- g(t)
The Correct Answer and Explanation is:
General Answer
The left-hand limit of g(t)g(t) as t→2−t \to 2^-: limt→2−g(t)\lim_{t \to 2^-} g(t)
is the value that the function g(t)g(t) approaches from the left side of 2. You examine the y-values that g(t)g(t) gets close to as tt gets closer and closer to 2 from values less than 2 (such as 1.9, 1.99, 1.999, etc.).
- If the graph smoothly approaches a specific y-value from the left, that value is the left-hand limit.
- If the graph jumps or has a discontinuity at t=2t = 2, the limit may still exist from the left, even if the right-hand limit or actual function value at t=2t=2 is different.
- If the graph goes off to infinity, or oscillates, or has a vertical asymptote, then the limit does not exist.
Explanation
In calculus, limits help us understand the behavior of a function as its input approaches a certain value. Specifically, the left-hand limit limt→2−g(t)\lim_{t \to 2^-} g(t) describes how the function behaves as tt approaches 2 from values less than 2. We only consider values of tt that are slightly less than 2 and observe what y-value g(t)g(t) approaches.
To determine this limit from a graph, we trace the curve of the function as it moves toward t=2t = 2 from the left. If the function approaches a single, definite y-value (say, 5), then the left-hand limit exists and equals that value. This is true even if the function is not defined at t=2t = 2, or if it has a different value at that point. Limits concern only the approach, not the actual point.
However, if the graph of the function jumps suddenly to another y-value at t=2t = 2, the left-hand limit may still exist, as long as from the left, the y-values are settling on a specific number. If instead the graph shoots up or down to infinity, or oscillates wildly without settling, then the limit does not exist.
In short, the limit limt→2−g(t)\lim_{t \to 2^-} g(t) exists if and only if the graph of g(t)g(t) approaches a specific, finite y-value from the left. You should use the graph to visually confirm what value, if any, is approached as tt comes from the left.
