Question

A)Using Microsoft Excel or Word, sketch or draw the general shape of the graph for the function

(x) = { x²+9 , x < 2
x²-3 , x ≥ 2
b)For the above function f(x), calculate the left-hand limit.
c) For the above function f(x), calculate the right-hand limit.
d)For the above function f(x), calculate the exact limit.
e)Does the limit as x tends to 2 exist?

Answer 1

To sketch the graph of the piecewise function f(x) and determine the limits as x approaches 2, one would draw two parabolas and examine the functional values as x approaches 2 from both sides, noting that the existence of the limit depends on the continuity at that point.

Step-by-step explanation:

The question asks about sketching the graph of a piecewise function f(x) and determining the limits as x approaches 2. To address the task:

a) One could sketch two different parabolas in Microsoft Excel or Word: x²+9 for x < 2, and x²-3 for x ≥ 2. b & c) To find the left-hand limit as x approaches 2, you would substitute x with values just smaller than 2 into x²+9. For the right-hand limit, you'd substitute x with values just greater than 2 into x²-3. Assuming the two parabolic pieces meet or approach the same point as x approaches 2 from both sides, the exact limit as x approaches 2 could be determined.

d) However, the exact limit as x approaches 2 does not exist if the functional values from the left and the right side do not match, indicating a discontinuity or jump in the function at that point.

e) Whether the limit exists or not depends on the continuity of the function at x = 2. If the function has the same value approaching from both sides, then the limit exists and is equal to that value.

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