http://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1010&context=books Acti

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http://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1010&context=books

Activity 2.23

Activitv 2.23. In this activity, we reason graphically from the following figure to evaluate limits of ratios of functions about which some information is known. 4 1234 Figure 2.21: Three graphs referenced in the questions of Activity 2.23. (a) Use the left-hand graph to determine the values of,f(2),(2), g(2), and ,(2) Then, evaluate im x-2 g(x) (b) Use the middle graph to find p(2), p'(2), q(2), and q'(2). Then, determine the 2.8. USING DERIVATIVES TO EVALUATE LIMITS value of lim x-2 q(x) (c) Use the right-hand graph to compute r(2), r'(2), s(2), s'(2). Explain why you cannot determine the exact value of

Explanation / Answer

(a)

f(2)=0,g(2)=0

tangent to f(x) at x =2 passes through (1,-0.5),(3,0.5)

f '(2)=(0.5-(-0.5))/(3-1)

f '(2)=1/2

tangent to g(x) at x =2 passes through (1.5,-2),(2.5,2)

g'(2)=(2-(-2))/(2.5-1.5)

g'(2)=4/1

g'(2)=4

by l hospital's rule limx->2(f(x))/(g(x))=limx->2(f '(x))/(g'(x)) =(f '(2))/(g'(2))=(1/2)/(4/1)=(1/8)

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(b)

p(2)=1.5,q(2)=1.5

tangent to p(x) at x =2 passes through (2,1.5),(3,0.5)

p'(2)=(0.5-1.5)/(3-2)

p'(2)=-1/1

p'(2)=-1

tangent to q(x) at x =2 is parallel to x axis

q'(2)=0

limx->2(p(x))/(q(x))=(p(2))/(q(2))=1.5/1.5 =1

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(c)

r(2)=0, s(2)=0

tangent to r(x) at x =2 is parallel to x axis ,tangent to s(x) at x =2 is parallel to x axis

r'(2)=0, s'(2)=0

limx->2(r(x))/(s(x)) is undefined

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