2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
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1 Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the function s end behavior, or the way it behaves on the far right-hand and far left-hand sides of the graph. For example, consider the function f(x) /x, graphed below: Note that, on the far right-hand side of the graph, the function values get very close to 0. Similarly, on the far left-hand side of the graph, f(x) 0. The concept of a function s end behavior is important enough that we give it a name: Definitions -2. Suppose that f(x) is defined on (a, ) (for definition ) or (, b) (for definition 2).. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large x values, we say that f(x) L. 2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large negative x values, we say that f(x) L. x Using the definitions above, we can now describe the behavior we observed in f(x) /x: x 0 and x x 0.
2 We must be careful to understanding the concept of its at infinity correctly. We are not plugging in in for x, since is not a number. Instead, we are looking for a trend in f(x) as its inputs become arbitrarily large (or large negative). So when we say that x 0, what we mean is that, as the x inputs grow large, /x trends towards 0. Notice in the graph above that y 0 is a horizontal asymptote of f(x) /x. Given that /x 0, it is no coincidence that f(x) has y 0 as an asymptote. In fact, horizontal asymptotes are defined in terms of its at infinity: Definition 3. If either of f(x) L or f(x) L, x we say that the line y L is a horizontal asymptote of f(x). Vertical Asymptotes vs. Horizontal Asymptotes It is important to remember the difference between vertical and horizontal asymptotes: the line x a is a vertical asymptote of f(x) if f(x) ± or x a + f(x) ± ; x a 2
3 on the other hand, the line y L is a horizontal asymptote of f(x) if f(x) L or The function f(x) /(x ) is graphed below: f(x) L. x The vertical asymptote x above appears because x + the horizontal asymptote y 2 appears because x ; x 2. Example. Evaluate the it 4. Since f(x) 4 is a constant function, it trends towards 4 regardless of x input: 3
4 So 4 4. Example. Evaluate the it x x2. As the x inputs for f(x) become increasingly large negative numbers, will become extremely large: x f(x) Since the outputs trend towards as the inputs x, we see that x x2. Example. Evaluate the it cos x. To understand this it, we must recall an important fact we learned earlier in this class: f(g(x)) f( g(x) ). x a x a 4
5 In other words, we can rewrite as so We already know that cos x cos ( ). x x 0, x cos( cos 0. The function is graphed below; it is clear that y is a horizontal asymptote for f(x): ) x Rules for Calculating Limits at Infinity The following rules can help us quickly calculate its at infinity for a few different functions:. If f(x) c is a constant function, then f(x) c. 2. If f(x) is a polynomial, then either sign of the highest term. f(x) or f(x), depending on the 3. If f(x) /x r, then f(x) 0 and f(x) 0. x 4. If f(x) results in an expression of the form or, the result is inconclusive, and we will need to do more work to evaluate the it. following techniques may be helpful: The 5
6 (a) If f(x) is a rational function, find the highest power of x that occurs in the denominator and divide through by this power of x; then try evaluating the it of the new function. (b) If the form appeared, try to rewrite the function by adding fractions or multiplying by conjugates, then reevaluate the it. Example. Find 8 x Notice that the it results in the form /. We can use the simple algebraic trick above to help us out here; find the term with the highest power of x appearing in the denominator of the fraction, and divide both numerator and denominator through by this power of x. In this case, 3 has the highest power of the terms in the denominator, so we will divide through by. We rewrite the fraction as 8 x x Next, we can use properties of its to help us out: since we can evaluate the it as follows: f(x) f(x) g(x) g(x), 8 x x ( 8 x + ) ( 3 ) x + 3 So 8 x + 3 6, + 3 which means that the function has the line y 6 as a horizontal asymptote, as the graph below indicates: 6
7 Example. Evaluate + 3x + x. Let s break the it into two pieces: we know that + 3x + x + 3x so we ll think about the its of the two terms separately. Using the rule f(g(x)) f( g(x) ), x a x a we can rewrite as Since + 3x x2 + 3x. x2 + 3x, the inputs of the radical are growning without bound; thus + x, Using the same trick, we see that + 3x. + x ; thus our original it + 3x + x 7
8 produces the form. We don t know what this means, so we must rewrite our function in order to better understand the it. Let s try multiplying by conjugates: + 3x + x ( + 3x + x) + 3x + + x + 3x + + x ( + 3x + x) ( + 3x + + x) + 3x + + x (x2 + 3x) ( + x) + 3x + + x + 3x x + 3x + + x + 3x + + x. as Now let s try evaluating the it again: we have rewritten + 3x + x + 3x + + x. Since the second it above is a rational function, we ll divide numerator and denominator through by the term from the denominator with the highest power of x; this is clearly x. So we rewrite the it: + 3x + + x x + 3x + + x 2 + 3x + + x We see that y is a horizontal asymptote of the function: 8
9 9
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