Section 1.3 Evaluating Limits Analytically

Transcription

1 Section 1.3 Evaluating Limits Analytically

2 Welcome to BC Calculus Wed August 7 th Use Graph and Table to find limits:

3 BC: Due Monday night (extension) CalcChat WolframAlpha Bin not black hole

4 Epsilon Delta

5 Strategies for Finding Limits To find limits analytically, try the following: 1. Direct Substitution (Try this FIRST). If Direct Substitution fails, then rewrite then find a function that is equivalent to the original function except at one point. Then use Direct Substitution. Methods for this include Factoring/Dividing Out Technique Rationalize Numerator/Denominator Eliminating Embedded Denominators Trigonometric Identities Legal Creativity

6 Direct Substitution One of the easiest and most useful ways to evaluate a limit analytically is direct substitution (substitution and evaluation): Example: If you can plug c into f(x) and generate a real number answer in the range of f(x), that generally implies that the limit exists (assuming f(x) is continuous at c). x 3 lim x 3 8 Always check for substitution first. The slides that follow investigate why Direct Substitution is valid.

7 Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Constant Function Limit of x lim() fxl xc lim() gx K xc lim bb xc lim xc xc Limit of a Power of x Scalar Multiple lim xc n xc n lim() bfxbl xc

8 Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Sum Difference Product Quotient lim() fxl xc lim() gx K xc lim()() fxgxlk xc lim()() fxgxlk xc fx () L lim, K 0 xc() gx K Power lim() n n fxl xc

9 Example lim f x 4 Let and. Find the following limits. x5 lim g x x5 1. lim fx5gx. lim fxgx x5 fx+lim 5gx fxlim x5 x5 fx+5lim x5 x5 gx x5 x5 x5 4 8 gx 3. lim x5 fx gx lim f x x5 lim x5 4 gx

10 Direct Substitution Example 53 Evaluatelim931 xxx x limlim9lim3lim1 5 3 xxx x x x x Sum/Difference Property 5 3 lim9lim3lim1 x x x x x x Multiple and Constant Properties 5 3 lim9lim3lim1 x x x x x x Power Property Limit of x Property

11 Direct Substitution Direct substitution is a valid analytical method to evaluate the following limits. If p is a polynomial function and c is a real number, then: lim() px pc () xc If r is a rational function given by r(x) = p(x)/q(x), and c is a real number, then pc () lim()() rxrc,()0 qc xc qc () If a radical function where n is a positive integer. The following limit is valid for all c if n is odd and only c>0 when n is even: lim n xc n xc

12 Direct Substitution Direct substitution is a valid analytical method to evaluate the following limits. If the f and g are functions such thatlim() gxlandfxfl lim()() xc xl Then the limit of the composition is: lim(()lim()() fgxf gxfl xc xc If c is a real number in the domain of a trigonometric function then: limsin xc sin xc limtan xc tan xc limsec xc sec xc limcos xc cos xc limcot xc cot xc limcsc xc csc xc

13 Example 1 f g Find lim g f x if 5 10, f 18 3, g 5 18, and x5 if f and g are continuous functions. 10 lim g f x g lim f ( x) x5 x5 g f (5) g 10

14 Example x Evaluate lim xx Direct Substitution can be used since the function is well defined at x=3 For what value(s) of x can the limit not be evaluated using direct substitution? At x=-6 since it makes the denominator 0: 0 66

15 Evaluate the limit analytically: x Indeterminate Form 44 xx lim x x An example of an indeterminate form because the limit can currently not be determined. 1/0 is NOT indeterminate. Note: If direct substitution results in 0/0 (or other indeterminates: /, x0, - ), the limit probably exists. Often limits can not be evaluated at a value using Direct Substitution. If this is the case, try to find another function that agrees with the original function except at the point in question. In other words x 4x4 How can we simplify:? x x

16 Example 1 Evaluate the limit analytically: x 44 xx lim x x At first Direct Substitution fails because x= results in 0/0. (Remember that this means the limit probably exists.) x xx 1 lim xx x xx 1 lim xx x lim x x Factor the numerator and denominator Cancel common factors This function is equivalent to the original function except at x= Direct substitution

17 Example Evaluate the limit analytically: lim y y y y y 4 y y y y lim y y lim y y lim 1 y y Rationalize the numerator Cancel common factors Direct substitution

18 Example 3 Evaluate the limit analytically: lim x x x Cancel the denominators of the fractions in the numerator 3x 3x lim x x x x x 3 x x lim x x 3 x x lim 1 3 x x lim x If the subtraction is backwards, Factoring a negative 1 to flip the signs Cancel common factors Direct substitution

19 Example 4 Evaluate the limit analytically: lim h 0 Expand the the expression to see if anything cancels h 5 5 h lim h0 h5 lim h0 h55 h h 10h55 h h 10h h hh10 h h0 lim h0 lim lim h0 h 10 Factor to see if anything cancels Direct substitution

20 Example 5 Evaluate the limit analytically: lim x 4 Rewrite the tangent function using cosine and sine 1tan x sin xcosx lim x 4 lim x 4 lim lim x 4 x 1 4 cos sin x cos x sin xcosx sin xcosx cosx cosx cosxsin x sin xcosxcosx sin xcosxcosx 1 cosx Eliminate the embedded fraction If the subtraction is backwards, Factoring a negative 1 to flip the signs Direct substitution

21 Two Freebie Limits The following limits can be assumed to be true (they will be proven later in the year) to assist in finding other limits: sin x lim 1 x 0 x 1cos x lim 0 x 0 x Use the identities to help with these limits. They are located on the first page of your textbook.

22 Example 1 Evaluate the limit analytically: lim x0 sin3 x 5x 3 3 lim x0 3sin3 x 53 x If 3x is the input of the sine function then 3x needs to be in the denominator 3 sin3x 5 3x x0 3 lim sin3 x 5 3x x0 lim Isolate the freebie Scalar Multiple Property Assumed Trig Limit 3 5

23 Evaluate the limit analytically: lim x0 1cosx x sin x Try multiplying by the reciprocal 1cosx 1cosx Example lim x0 lim x0 lim x0 lim x0 lim x0 1cosxcosxcos x x sin x1cosx 1cos x x sin x 1cosx sin x x sin x 1cosx sin x x1cosx sin x x 1 1 1cos0 1 lim x0 1 1cosx Use the Trigonometry Laws Split up the limits A freebie limit and Direct substitution