Limits and Continuity in AP Calculus AB/BC

What is the primary purpose of using limits in calculus?

• To calculate the total area under a curve

• To calculate the average rate of change over an interval

• To find the instantaneous rate of change at a specific point

• To determine the highest and lowest points on a graph

How can you estimate the average rate of change using a table of values?

• By finding the slope of the tangent line at a given point

• By calculating the difference in y-values divided by the difference in x-values

• By applying L'Hopital's rule to the given values

• By using the Squeeze Theorem on the table's values

What does the conjugate method help to solve in limit problems?

• Indeterminate forms like 0/0 by rationalizing the numerator or denominator

• Limits involving trigonometric functions

• Limits that require factoring polynomials

• Limits that can be directly substituted

Which of the following is true about the Squeeze Theorem?

• It applies only to limits at infinity

• It is used to calculate the area between curves

• It requires the use of L'Hopital's rule for verification

• It is primarily used for solving limits that involve trigonometric functions

When is L'Hopital's rule applicable for solving limits?

• When the limit results in an indeterminate form like 0/0 or ∞/∞

• For all types of limits without exception

• Only for limits involving exponential functions

• Only when the limit cannot be solved by direct substitution

What theorem is used to evaluate a limit by comparing it to two other functions that squeeze it?

• Limit Comparison Theorem

• Squeeze Theorem

• Intermediate Value Theorem

• Fundamental Theorem of Calculus

Which type of discontinuity occurs at a vertical asymptote?

• Removable Discontinuity

• Point Discontinuity

• Jump Discontinuity

• Infinite Discontinuity

What is the condition for a function to be continuous at a point x = c?

• The function must be differentiable at x = c

• The left-hand limit and right-hand limit at x = c must be finite

• The function must be defined at x = c, the limit as x approaches c exists, and both are equal

• The function must have a horizontal asymptote at x = c

How can a removable discontinuity be fixed in a rational function?

• By adding a constant to the function

• By defining the function as a piecewise function

• By multiplying the function by x

• By taking the derivative of the function

What does the Intermediate Value Theorem guarantee about a function on an interval [a, b]?

• The function will have at least one root in the interval if it is continuous

• The function will have no discontinuities in the interval

• The function's limit as x approaches b will be the same as its value at a

• The function will be differentiable on the interval