Study Sheet - Midterm 1 (Functions, Limits, Asymptotes & Continuity) Solutions - 1 1.1 Solution for midterm1 part1 Some formulas Ex(1 For

# Study Sheet - Midterm 1 (Functions, Limits, Asymptotes & Continuity) Solutions

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1 Solution for midterm1, part1 1.1 Some formulas Ex(1) : For x -intercept, set y = x 2 9 - x 2 = 0, solve for x , we have x = 0 , ± 3. So the x -intercepts are (0 , 0) , ( - 3 , 0) , (3 , 0). For y -intercept, set x = 0. We have y = 0. So the y -intercept is (0 , 0). Ex(2) : Use the formula for circle: ( x - 2) 2 + ( y - 3) 2 = 5 2 . Ex(3) : Use slope-intercept formula, the equation for the line is y = - x + 3. Next, set y from both functions equal to each other,i.e. x 2 +1 = - x +3, and solve for x . we have ( x + 2)( x - 1) = 0 x = - 2 , 1. Plug them back into to either function. The intercepts are ( - 2 , 5) , (1 , 2). 1.2 Function Ex(1) : f ( g ( x )) = 1 x 2 - 1 , its domain is obtained by setting x 2 - 1 6 = 0. Thus, the domain of f ( g ( x )) is x 6 = ± 1. g ( f ( x )) = ( 1 x ) 2 - 1, its domain is x 6 = 0. Ex(2) : Set y = x 2 - 4 and solve for x . Note x > 2, we have y 2 = x 2 - 4 x 2 = y 2 + 4 x = p y 2 + 4 The inverse of the original function is f - 1 ( x ) = x 2 + 4. 1.3 Limits Ex(1) : (a), lim x 2 x 4 = 2 4 ; (b), lim x 3 x + 1 = 4 = 2 (c), lim x →- 2 3 x +1 2 - x = lim x →- 2 3 x +1 lim x →- 2 2 - x = - 5 4 ; (d), lim x 3 x +1 - 1 x = lim x 3 ( x +1 - 1) lim x 3 ( x ) = 1 3 Ex(2) : (a), lim x →- 1 x 2 - 1 x +1 =lim x →- 1 ( x - 1)( x +1) x +1 =lim x →- 1 ( x - 1) = - 2 (1) 1
(b), lim x 1 x 3 - 1 x - 1 =lim x 1 ( x - 1)( x 2 + x +1) x - 1 =lim x 1 ( x 2 + x + 1) = 3 (2) (c), lim Δ x 0 2( x x
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