Algebraic Limits and Continuity OBJECTIVE Develop and use the Limit Properties to calculate limits. Determine whether a function is continuous at a point.

1.2 Algebraic Limits and Continuity LIMIT PROPERTIES: If and and c is any constant, then we have the following limit properties: L1. The limit of a constant is the constant.

1.2 Algebraic Limits and Continuity LIMIT PROPERTIES (continued): L2. The limit of a power is the power of that limit, and the limit of a root is the root of that limit. where m is any integer, L ≠ 0 if m is negative, where n ≥ 2 and L ≥ 0 when n is even.

1.2 Algebraic Limits and Continuity LIMIT PROPERTIES (continued): L3. The limit of a sum or difference is the sum or difference of the limits. L4. The limit of a product is the product of the limits.

1.2 Algebraic Limits and Continuity LIMIT PROPERTIES (concluded): L5. The limit of a quotient is the quotient of the limits. L6. The limit of a constant times a function is the constant times the limit of the function.

1.2 Algebraic Limits and Continuity Example 1: Use the limit properties to find We know that By Limit Property L4,

1.2 Algebraic Limits and Continuity Example 1 (concluded): By Limit Property L6, By Limit Property L1, Thus, using Limit Property L3, we have

1.2 Algebraic Limits and Continuity THEOREM ON LIMITS OF RATIONAL FUNCTIONS For any rational function F, with a in the domain of F,

1.2 Algebraic Limits and Continuity Example 2: Find The Theorem on Limits of Rational Functions and Limit Property L2 tell us that we can substitute to find the limit:

1.2 Algebraic Limits and Continuity Quick Check 1 Find the following limits and note the Limit Property you use at each step: a.) b.) c.)

1.2 Algebraic Limits and Continuity Quick Check Solution a.) We know that the 1.) Limit Property L4 2.) Limit Property L6 3.) Limit Property L4 4.) Limit Property L6 5.) Limit Property L1 6.) Combining steps 2.), 4.), and 5.) we get Limit Property L3

1.2 Algebraic Limits and Continuity Quick Check 1 solution b.) We know that 1.) Limit Properties L4 and L6 2.) Limit Property L6 3.) Limit Property L1 4.) Combine above steps: Limit Property L3 5.) Limit Property L6 6.) Limit Property L1 7.) Combine above steps: Limit Property L3 8.) Combine steps 4.) and 7.) Limit Property L5

1.2 Algebraic Limits and Continuity Quick Check 1 solution c.) We know that 1.) Limit Property L1 2.) Limit Properties L4 and L6 3.) Combine above steps: Limit Property L3 4.) Using step 3.) Limit Property L2

1.2 Algebraic Limits and Continuity Example 3: Find Note that the Theorem on Limits of Rational Functions does not immediately apply because –3 is not in the domain of However, if we simplify first, the result can be evaluated at x = –3

1.2 Algebraic Limits and Continuity Example 3 (concluded): This means that the limits exist as x approaches -3, but the actual point does not (from previous slide).

1.2 Algebraic Limits and Continuity DEFINITION: A function f is continuous at x = a if: 1) exists, (The output at a exists.) 2) exists, (The limit as exists.) 3) (The limit is the same as the output.) A function is continuous over an interval I if it is continuous at each point a in I. If f is not continuous at x = a, we say that f is discontinuous, or has discontinuity, at x = a.

1.2 Algebraic Limits and Continuity Example 4: Is the function f given by continuous at x = 3? Why or why not? 1) 2) By the Theorem on Limits of Rational Functions, 3) Since f is continuous at x = 3.

1.2 Algebraic Limits and Continuity Example 5: Is the function g given by continuous at x = –2? Why or why not? 1) 2) To find the limit, we look at left and right-side limits.

1.2 Algebraic Limits and Continuity Example 8 (concluded): 3) Since we see that the does not exist. Therefore, g is not continuous at x = –2.

1.2 Algebraic Limits and Continuity Quick Check 2 Let Is g continuous at ? Why or why not? 1.) 2.) To find the limit, we look at both the left-hand and right-hand limits: Left-hand: Right-hand: Since we see that does not exist. Therefore g is not continuous at

1.2 Algebraic Limits and Continuity Quick Check 3a Let Is h continuous at Why or why not? In order for to continuous, . So lets start by finding . So the . However, , and thus . Therefore is not continuous at .

1.2 Algebraic Limits and Continuity Quick Check 3b Let Determine c such that p is continuous at In order for to be continuous at , . So if we find , we can determine what c is. Let’s find : So . Therefore, in order for p to be continuous at , .

1.2 Algebraic Limits and Continuity Section Summary For a rational function for which is in the domain, the limit as approaches can be found by direct evaluation of the function at If direct evaluation leads to the indeterminate form , the limit may still exist: algebraic simplification and/or a table and graph are used to find the limit. Informally, a function is continuous if its graph can be sketched without lifting the pencil off the paper.

1.2 Algebraic Limits and Continuity Section Summary Continued Formally, a function is continuous at if: The function value exists The limit as x approaches a exists The function value and the limit are equal This can be summarized as If any part of the continuity definition fails, then the function is discontinuous at