AP Calculus AB Exam Question:
Find the limit of the function f(x) = 3x^2 - 2x + 1 as x approaches 2.
Answer:
To find the limit of the function f(x) = 3x^2 - 2x + 1 as x approaches 2, we substitute x = 2 into the function and evaluate the result.
Step 1: Substitute x = 2 into the function:
f(2) = 3(2)^2 - 2(2) + 1 = 12 - 4 + 1 = 9
Therefore, the value of the function at x = 2 is 9.
Step 2: Since we only substituted x = 2 into the function, we cannot directly conclude that the limit of the function as x approaches 2 is 9. We need to investigate further because a limit is determined by the behavior of a function as x approaches a certain value.
Step 3: To examine the behavior of the function as x approaches 2, we can either construct a table of values or graph the function.
Constructing a table of values for the function:
| x | f(x) |
|---|---|
| 1 | 2 |
| 1.5 | 3.25 |
| 1.9 | 5.19 |
| 1.99 | 8.0699 |
| 1.999 | 8.00699 |
| 2 | 9 |
| 2.001 | 9.00699 |
| 2.01 | 9.0699 |
| 2.1 | 9.19 |
| 2.5 | 10.75 |
| 3 | 16 |
Step 4: By examining the table of values and the graph of the function, we can see that as x approaches 2, the values of f(x) gets arbitrarily close to 9.
Step 5: Therefore, we can conclude that the limit of the function f(x) = 3x^2 - 2x + 1 as x approaches 2 is 9.
In mathematical notation, we write:
lim(x->2) (3x^2 - 2x + 1) = 9