Math 222 – Practice Exam 1 w Solutions 2011 – Calculus 2

Math 222 Calculus 2 Fall 2008 Exam 1 Solutions(6) (20 Points) Evaluate the following limits. If you use L’Hospital’s Rule, show where youuse it and explain what type of limit you are using it on.(a) limx→0sin(2x 3 )sin(3x 3 )(7 points) By L’Hospital’s Rule for a 0 0-type indeterminate form,sin(2x 3 )limx→0 sin(3x 3 )L ′ H=6x 2 cos(2x 3 )limx→0 9x 2 cos(3x 3 ) = lim (6)(1)x→0 (9)(1) = 2 3 .(b) limx→0 + √ xln(x)(6 points) limx→0 + √ xln(x) is an indeterminate form of the type (0)(−∞), which we makeinto a ∞ ln(x)∞-type by writing it as limx→0 + x −1/2lim −2x 1/2 = 0x→0 +L ′ H=1/xlimx→0 + −1= limx−3/22xx→0 + −2x 3/2=(c) limx→0(1 + x + 2x2 ) 1/x.(7 points) Let y = ( 1 + x + 2x 2) 1/x ln(1 + x + 2x 2 )so ln(y) = . Thenxln(1 + x + 2x 2 )lim ln(y) = limx→0 x→0 xis a type 0 0indeterminate form. L’Hospital’s Rule givesln(1 + x + 2x 2 )limx→0 xL ′ H=limx→01+4x1+x+2x 21= 1so y → e 1 = e is the limit we seek.