MAT137Y1 Midterm: 2008 Test 1 solution

• School: UTSG
• Department: Mathematics
• Course: MAT137Y1
• Professor: N/ A

Document Summary

Mat 137y, 2008-2009 winter session, solutions to term test 1: evaluate the following limits. (do not prove them using the formal de nition of limit. ) (10%) (i) lim x 0. Multiplying top and bottom by 1 + cosx, we have xsinx. 1 cosx xsinx(1 + cosx) (1 cosx)(1 + cosx) lim x 0 xsinx(1 + cosx) xsinx(1 + cosx) (1 + cosx) = 1 2 = 2. sin2 x. = lim x 0 x sinx (10%) (ii) lim x 3. Here we multiply top and bottom by the conjugates of both expressions to get x2 7 (cid:32) . [(x + 6) 9][ x2 7] (3 x)(4 + x)( x + 6 + 3) (x 3)( = lim x 3 x2 7 lim x 3 x2 7 x2 7. 5 x + x 3 x + 6 + 3) (4 + x)( 2. (10%) (i) find all solutions in the interval [0,2 ) that satisfy the equation 2sin3x 1 = 0.