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Learn to find limits by reading graphs and understanding visual behavior
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When you have a graph of a function, you can find limits by observing where the function is heading as approaches a specific value.
To find from a graph:
Given the graph of below, evaluate the following limits:
a) b) c) d)
| Section | Format | Questions | Time | Weight | Calculator |
|---|---|---|---|---|---|
| Multiple Choice (No Calculator) | MCQ | 30 | 60 min | 33.3% | ๐ซ |
| Multiple Choice (Calculator) | MCQ | 15 | 45 min | 16.7% | โ |
| Free Response (Calculator) | FRQ | 2 | 30 min | 16.7% | โ |
| Free Response (No Calculator) | FRQ | 4 | 60 min | 33.3% | ๐ซ |
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See how this math is used in the real world
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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If the function is continuous at , the limit equals the function value:
Even if there's a hole at , the limit can still exist. The limit is where the hole "should be" filled.
If the function jumps from one value to another at , the left and right limits differ, so the limit does not exist.
If the function grows without bound as , we say the limit is infinite (or doesn't exist as a real number).
Always check BOTH sides of the point. If they don't agree, the limit does not exist!
Assume the graph shows:
Solution:
Part (a): is the left-hand limit.
As approaches 2 from the left, approaches 3.
Part (b): is the right-hand limit.
As approaches 2 from the right, approaches 3.
Part (c): exists if and only if both one-sided limits exist and are equal.
Since and :
Part (d): is the actual function value at .
The solid dot is at , so:
Note: The limit equals 3, but the function value is 5. The function has a removable discontinuity at .