Explain the concept of limits.
Limits describe the behavior of a function as the input approaches a specific value. They are fundamental to calculus.
Explain instantaneous rate of change.
It's the rate of change at a single point, found by taking the limit of the average rate of change as the interval approaches zero.
Explain why limits matter.
Limits allow us to analyze function behavior at points where the standard rate of change formula would be undefined.
Explain the relationship between secant and tangent lines.
As the distance between two points on a curve approaches zero, the secant line approaches the tangent line.
Explain the connection between limits and derivatives.
The derivative of a function at a point is defined as the limit of the difference quotient as the change in x approaches zero.
How do you evaluate a limit graphically?
Examine the graph of the function as x approaches the specified value from both sides. If the y-value approaches the same value from both sides, that is the limit.
How do you evaluate a limit algebraically?
Try direct substitution first. If it results in an indeterminate form (e.g., 0/0), try factoring, rationalizing, or other algebraic manipulations to simplify the expression before evaluating the limit.
How do you handle limits that result in indeterminate forms?
Use algebraic techniques like factoring, rationalizing, or L'Hôpital's Rule (if applicable) to simplify the expression before evaluating the limit.
What does the graph of a function tell us about its limit as x approaches a certain value?
The graph shows the y-value that the function approaches as x gets closer and closer to the specified value. Discontinuities can affect the existence of the limit.
