Mastering Limits: The Foundation of Calculus

1. In the tangent-slope idea, why do we use a secant line instead of trying to compute the tangent slope directly at the single point?

2. As point Q approaches point P (without becoming equal), what happens to the secant line compared to the tangent line?

3. Why is it not allowed to set Q = P directly in the secant approach?

4. Which condition is necessary for the two-sided limit lim(x→a) f(x) to exist?

5. Suppose the limit from the right and from the left approach different values. What is the value of the two-sided limit?

6. In one-sided limit notation, what does the superscript '+' indicate?

7. A limit does NOT require the function to have a specific value at x = a. What does it focus on instead?

8. When computing a limit numerically with a table, what should you do to test the left-hand and right-hand behaviors?

9. If f(x) approaches +Infinity as x approaches a from both sides, which statement is correct?

10. In the example f(x) = 1/x as x approaches 0, what happens to the right-hand and left-hand limits?

1. The core idea of a limit is to see what f(x) approaches as x gets really close to a value _____ (without necessarily reaching it).

2. A tangent line at a point can be found by taking the slope of secant lines as Q gets _____ to P (but never equals P).

3. For a two-sided limit lim(x→a) f(x) to exist, the left-hand limit and right-hand limit must approach the same _____ value.

4. In one-sided notation, a superscript _____ means approaching from the right.

5. In numerical limit tables, to find a two-sided limit you should evaluate f(x) at x-values close to a coming from both the _____ and the _____ sides.