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##### Definition of Limit, Properties of Limits

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**Definition of Limit,Properties of Limits**Section 2.1a**Let’s start with an exploration…**What are the values of the function given below as x approaches 0??? First, graph the function in the window by Now, look at a table, with TblStart = –0.3 and Tbl = 0.1 –.3 .98507 –.2 .99335 –.1 .99833 0 ERROR .1 .99833 .2 .99335 .3 .99507**Let’s start with an exploration…**What are the values of the function given below as x approaches 0??? –.3 .98507 –.2 .99335 –.1 .99833 0 ERROR .1 .99833 .2 .99335 .3 .99507 What do these steps suggest? Note: We cannot simply substitute x = 0 into the function, because we’d be dividing by zero…………we need another method…**Definition: Limit**Let c and L be real numbers. The function f has limit L as x approaches cif, given any positive number , there is a positive number such that for all x, We write which is read, “the limit of f of x as x approaches c equals L.” The notation means that the values of f(x) of the function f approach or equal L as the values of x approach (but do not equal ) the number c…**Definition: Limit**Let c and L be real numbers. The function f has limit L as x approaches cif, given any positive number , there is a positive number such that for all x, We write As suggested in our opening exploration:**Important note: The existence of a limit as x c never**depends on how the function may or may not be defined at c.**Definition: Limit**Let c and L be real numbers. The function f has limit L as x approaches cif, given any positive number , there is a positive number such that for all x, We write Find each of the following limits:**Properties of Limits**If L, M, c, and k are real numbers and and , then • Sum Rule – The limit of the sum of two functions is the sum of • their limits: 2. Difference Rule – The limit of the difference of two functions is the difference of their limits:**Properties of Limits**If L, M, c, and k are real numbers and and , then 3. Product Rule – The limit of a product of two functions is the product of their limits: 4. Constant Multiple Rule – The limit of a constant times a function is the constant times the limit of the function:**Properties of Limits**If L, M, c, and k are real numbers and and , then 5. Quotient Rule – The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero:**Properties of Limits**If L, M, c, and k are real numbers and and , then 6. Power Rule – If r and s are integers, , then provided that is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number.**Guided Practice**Find each of the following limits. (a) Notice anything? (b)**Theorem:**Polynomial and Rational Functions 1. If is any polynomial function and c is any real number, then 2. If and are polynomials and c is any real number, then provided that**Guided Practice**Find each of the following limits. (a) (b)**Guided Practice**Explain why you cannot use substitution to determine the given limits. Find the limit if it exists. Cannot use substitution b/c the expression is not defined at x = 0. Since becomes arbitrarily large as x approaches 0 from either side, there is no (finite) limit. Can we support this reasoning graphically???**Guided Practice**Explain why you cannot use substitution to determine the given limits. Find the limit if it exists. Cannot use substitution b/c the expression is not defined at x = 0. A key rule for evaluating limits: If substitution will not work directly, use algebra techniques to re-write the expression so that substitution will work!!!**Guided Practice**Explain why you cannot use substitution to determine the given limits. Find the limit if it exists. Cannot use substitution b/c the expression is not defined at x = 0. Support graphically???**Guided Practice**Determine the given limits algebraically. Support graphically.**Guided Practice**Determine the given limits algebraically. Support graphically.**Guided Practice**Determine the given limits algebraically. Support graphically.