IndietroBusiness Calculus Exam 1 Study Guide: Step-by-Step Guidance
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Q4. Use the graph of below to find the following:
Background
Topic: Graphical Analysis of Functions and Limits
This question tests your ability to interpret a graph, evaluate function values, and analyze limits and discontinuities. You will use the graph to answer questions about at specific points and as approaches certain values.
Key Terms and Concepts:
Function Value: is the y-value of the function at .
Limit: is the value the function approaches as gets close to .
Discontinuity: A point where the function is not continuous (jump, hole, or asymptote).
One-sided Limits: (from the left), (from the right).
Step-by-Step Guidance
To find for a given value, locate on the graph and identify the corresponding y-value. If there is a filled dot, that is the function value; if not, the function may be undefined at that point.
For limits as approaches a value, observe the behavior of the graph as gets close to that value from both sides. If the left and right sides approach the same y-value, the limit exists.
For one-sided limits, focus only on the graph as approaches from the left () or right ().
To determine discontinuities, look for jumps, holes, or breaks in the graph. These are points where the function is not continuous.
For limits at infinity ( or ), observe the end behavior of the graph as increases or decreases without bound.
Try solving on your own before revealing the answer!
Final Answers:
a)
b)
c)
d)
e)
f)
g)
h) does not exist (DNE)
i)
j)
k)
l)
m)
n) does not exist (DNE)
o)
p) The function is discontinuous at
Each answer is based on careful reading of the graph, noting filled and open dots, jumps, and the behavior as approaches the specified values.
