Limits of Functions

Therefore, we can also find the limit of: For the sequence: We found its limit by evaluating: = 1 Therefore, we can also find the limit of: Evaluate: = 1 = 1

The Limit as x approaches a Real Number c: To determine what happens to a function as it approaches a real number c, we have to consider two possibilities: 1. which is read as “the limit of f(x) as x approaches c from the right.” 2. which is read as “the limit of f(x) as x approaches c from the left.”

Using the graph of f(x) below, find: = -2 = 5

Using the graph of f(x) below, find: = -1 = 4

If , describe the behavior of f(x) near x = 2. Solution. Find: and In general: exists, if and only if and exist and agree.

Continuous Functions A function is continuous if its graph can be drawn without lifting your pencil. The formal definition is that a function is continuous if: This means that are three conditions for a function to be continuous at x = c. 1. must exist 2. f(c) must exist 3. These two values must be equal

Determine whether the following function is continuous Determine whether the following function is continuous. If it is discontinuous, state where any discontinuities occur. Answer: Discontinuous at x = 0 Determine values for a and b so that the function is continuous. Answer: a = 0, b = 5

Evaluating: 1. Use the quotient theorem which is: if both limits exist and 2. If and try the following techniques: a. Factor g(x) and f(x) and reduce to lowest terms. b. If f(x) or g(x) involve a square root, try multiplying both f(x) and g(x) by the conjugate of the square root expression. 3. If and then either statement below is true: a. The does not exist b. The or 4. If x is approaching infinity or negative infinity, divide the numerator and the denominator by the highest power of x in the denominator.

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