Math 1314 Lesson 4 Limits Finding a it amounts to answering the following question: What is happening to the y-value of a function as the x-value approaches a specific target number? If the y-value is approaching a specific number, then we can state the it of the function as x gets close to the target number. Notation: is short for it. If we write (y value) as the x values approach. x it means, what s the it of f( Example 1: Given the graph of f( below. Find: A. x B. x 0 It does not matter whether or not the x value ever reaches the target number. It might, or it might not. Example : Given the following graph of a function f(, find. x 0 Lesson 4 Limits 1
Example 3: Given the following function, what is the it of the function as x approaches -? 5 3, 4, 9, When can a it fail to exist? We will look at two cases where a it fails to exist (note: there are more, but some are beyond the scope of this course). Case 1: The y value approaches one number from numbers smaller than the target number and it approaches a second number from numbers larger than the target number: Case : At the target number for the x-value, the graph of the function has an asymptote. For either of these two cases, we would say that the it as x approaches the target number does not exist. Lesson 4 Limits
The it formally defined: We say that a function f has it L as x approaches the target number a, written = L if the value f( can be made as close to the number L as we please by taking x sufficiently close to (but not equal to) a. Note that L is a single real number. Evaluating Limits Most often, we will not have a graph given, and will need to find a it by algebraic methods. Properties of its: Suppose = L and g( = M. Then, 1. r r r [ f ( ] = [] = L for any real number r.. cf ( = c = cl for any real number c. 3. ( ) ( ) ( ) ( ) f x ± g x = f x ± g x = L ± M 4. [ f ( g( ] = [][ g( ] = LM. f ( L 5. = =, provided M 0. g( g( M Substitution Example 4: Suppose f ( = x 3 4. Find. x 3 Lesson 4 Limits 3
Example 5: Evaluate: 3x 14 x 0 x + 6 3 Example 6: Evaluate: 5x x 8 Example 7: Evaluate: x + 8. x x 4 4 When substitution gives you a value in the form 0 k, where k is any non-zero real number the it DNE. Indeterminate Forms What do you do when substitution gives you the value 0 0? This is called an indeterminate form. It means that we are not done with the problem! We must try another method for evaluating the it!! Once we determine that a problem is in indeterminate form, we can use GGB to find the it. Command: it[<function>,<value>] Lesson 4 Limits 4
Example 8: Determine if the function given is indeterminate. If it is, use GGB to evaluate the it. x + 3x 10 x x 4 Example 9: Determine if the function given is indeterminate. If it is, use GGB to evaluate the it. x 5x + 6 x 3 x 3 Example 10: Determine if the function given is indeterminate. If it is, use GGB to evaluate the it. x 4 x x 4 Lesson 4 Limits 5
So far we have looked at problems where the target number is a specific real number. Sometimes we are interested in finding out what happens to our function as x increases (or decreases) without bound. Limits at Infinity Consider the function f x =. As the value of x get larger and larger, f( x + 1 ( approaches? x 10 1000 100,000 1,000,000 f( 1.98 1.99 1.999 1.999 We say that a function f( has the it L as x increases without bound (or as x approaches infinity), written = L, if f( can be made arbitrarily close to L by taking x large enough. We say that a function f( has the it L as x decreases without bound (or as x approaches negative infinity), written = L, if f( can be made arbitrarily close to L by taking x to x be negative and sufficiently large in absolute value. We can also find a it at infinity by looking at the graph of a function. Example 11: Evaluate: 6x 7 x Lesson 4 Limits 6
We can also find its at infinity algebraically or by recognizing the end behavior of a polynomial function. Even Degree Poly Odd Degree Poly Positive Negative Positive Negative Leading Coefficient Example 1: Evaluate: 3 A. ( 4x 7x + 5) x B. ( 3x 4 + 1) x Leading Coefficient 5 C. ( x 3 x Limits at infinity problems often involve rational expressions (fractions). The technique we can use to evaluate its at infinity is to divide every term in the numerator and the denominator of n the rational expression by x, where n is the highest power of x present in the denominator of the expression. Then we can apply this theorem: 1 Theorem: Suppose n > 0. Then = 0 n x 1 and = 0 x n x 1, provided n is defined. x However, often students prefer to just learn some rules for finding its at infinity. The highest power of the variable in a polynomial is called the degree of the polynomial. We can compare the degree of the numerator with the degree of the denominator and come up with some generalizations. If the degree of the numerator is smaller than the degree of the denominator, then f ( = 0. g( If the degree of the numerator is the same as the degree of the denominator, then you can f ( find by making a fraction from the leading coefficients of the numerator and g ( x ) denominator and then reducing to lowest terms. If the degree of the numerator is larger than the degree of the denominator, then it s best to work the problem by dividing each term by the highest power of x in the denominator and simplifying. You can then decide if the function approaches or, depending on the relative powers and the coefficients. Lesson 4 Limits 7
The notation, = indicates that, as the value of x increases, the value of the function increases without bound. This it does not exist, but the notation is more descriptive, so we will use it. Example 13: Evaluate: 5x + 3x + 4 4x x + 8 Example 14: Evaluate: 4x + 5 x 9x + 9 Example 15: Evaluate: x 4 x + 4 x x Example 16: Evaluate: 5 5x + x 4 x + 1 Lesson 4 Limits 8