MA 123 (Calculus I) Lecture 3: September 12, 2017 Section A2. Professor Jennifer Balakrishnan,

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1 What is on today Professor Jennifer Balakrishnan, 1 Techniques for computing limits Limit laws One-sided limits Other techniques Infinite limits and one-sided infinite limits Finding infinite limits analytically Vertical asymptotes Techniques for computing limits Briggs-Cochran-Gillett pp Limit laws Here are some very helpful limit laws which let us simplify the evaluation of many limits: Theorem 1 (Limit laws) Assume and lim exist. The following properties hold, where c is a real number, and m > 0, n > 0 are integers: 1. Sum: lim (f(x) + ) = lim f(x) + lim 2. Difference: lim (f(x) ) = lim f(x) lim 3. Constant multiple: lim(cf(x)) = c 4. Product: lim (f(x)) = 5. Quotient: lim ( ) f(x) = ( lim ) ( ) f(x) lim lim ( ) n 6. Power: lim(f(x)) n =, provided lim 0 7. Fractional power: lim(f(x)) n/m =, provided f(x) 0 for x near a, if m is even and n/m is reduced to lowest terms. 1

2 Note that the power law above is a special case of the fractional power law. Example 2 ( 2.3 Ex. 18, 20, 24) Assume = 8, lim = 3, and lim h(x) = 2. Compute the following limits and state the limit laws used to justify your computations. f(x) h(x) (f(x)h(x)) (c) lim 3 f(x) + 3 We record a few special cases (really, just applications of the limit laws above), limits of polynomial functions and rational functions: Theorem 3 Assume p and q are polynomials and a is constant. 1. Polynomial functions: lim p(x) = p(a) p(x) 2. Rational functions: lim q(x) = p(a), provided q(a) 0. q(a) Example 4 ( ,28) Evaluate the following limits: t 2 (t 2 + 5t + 7) t 3 3 t One-sided limits Theorem 5 (One-sided limit laws) Laws 1-6 in Theorem 1 hold with lim replaced with lim or lim +. Law 7 is modified as follows. Assume m > 0 and n > 0 are integers. 7. Fractional power =, provided f(x) 0, for x near a with x > a, +(f(x))n/m + if m is even and n/m is reduced to lowest terms. (f(x))n/m =, provided f(x) 0, for x near a with x < a, if m is even and n/m is reduced to lowest terms. 2

3 Example 6 ( 2.3 Ex. 34) Let 0 if x 5 f(x) = 25 x 2 if 5 < x < 5 3x if x 5. Compute the following limits or state that they do not exist. (a) (b) x 5 x 5 + (c) (d) x 5 x 5 (e) x 5 + (f) lim x 5 f(x) Example 7 ( 2.3 Based on Ex. 40) Compute the following limits or state that they do not exist. x 2 2x 3 x 4 x 3 x 3 x 2 2x 3 x 3 x 2 a 2 Example 8 ( 2.3 Ex. 50) Evaluate lim, where a > 0 is a real number. x a 1.3 Other techniques Example 9 ( 2.3 Ex. 55) Here we analyze the behavior of x sin 1 x 1. Show that x x sin 1 x for x 0. x near x = 0: 2. Illustrate the inequalities above with a graph. 3. Make a conjecture about lim x 0 x sin 1 x. Theorem 10 (Squeeze Theorem) Assume functions f, g, h satisfy f(x) h(x) for all values of x near a, except possibly at a. If then = lim h(x) = L, lim = L. 3

4 1.4 Infinite limits and one-sided infinite limits We will now look at functions for which values increase or decrease without a bound near a point. Example 11 ( 2.4 Ex. 10) Consider the graph of g in the figure. Analyze the following limits. 1. lim x 2 2. lim x lim x 2 4. lim x 4 5. lim x lim x 4 Definition 12 (Infinite Limits and One-sided Limits) Suppose f is defined for all x near a. If f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a we write = and we say that the limit of f(x) as x approaches is infinity. If f(x) is negative and grows arbitrarily large in magnitude for all x sufficiently close (but not equal) to a we write = and we say that the limit of f(x) as x approaches a is negative infinity. In both cases, the limit does not exist. Suppose now that f is defined for all x near a with x > a. If f(x) grows arbitrarily large for all x sufficiently close (but not equal) to a with x > a we write The one sided limits =, lim + analogously. =. + f(x) = and = are defined 4

5 1.4.1 Finding infinite limits analytically Example 13 ( 2.4 Ex. 24) Determine the following limits or state that they do not exist. 1. lim x 2 + x 3 5x 2 + 6x 2. lim x 2 x 3 5x 2 + 6x x 3 5x 2 + 6x 3. lim x 2 4. lim x 2 x 3 5x 2 + 6x 1.5 Vertical asymptotes Definition 14 (Vertical asymptotes) If = ±, = ±, or = ± then the line x = a is called a + vertical asymptote. Example 15 ( 2.4 Ex. 32 and 33) Find all vertical asymptotes x = a of the following functions. For each value of a determine, and f(x) = cos x x 2 + 2x 2. f(x) = x + 1 x 3 4x 2 + 4x 5

6.1 Polynomial Functions

6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and