ANSWERS AND EXPLANATIONS TO SECTION I
- C Use the double angle formula for sine, sin 2θ = 2 sin θ cos θ to rewrite the limit and then solve:
- B If we take the limit as x goes to , we get an indeterminate form , so let’s use L’Hôpital’s
Rule. We take the derivative of the numerator and the denominator and we get =. Now, when we take the limit we get .
3. A A removable discontinuity occurs when a rational expression has common factors in the
numerator and the denominator. The reduced function has the factor (x + 2) in the numeratorand denominator, hence there is a removable discontinuity when x = −2. The y-coordinate ofthe discontinuity is found by plugging x = −2 into the reduced function, f(x)=.Thus, the point where a removable discontinuity exists is (−2, −1).- C In (A), if you take the limit from both sides, you get: f(x) = 25 and f(x) = 26. The two
limits don’t match, so f(x) is not continuous at x = 3. In (B), first factor the numerator and thedenominator: g(x) = . Note that if you plug 3 into the numerator and thedenominator, you will get , so g(x) is not continuous at x = 3. In (C), first factor thenumerator and the denominator: h(x) = . Note that if you plug 3 into thenumerator and the denominator, you will get 0. The problem with the function is at x= −3,where it is not continuous. So h(x) is continuous x = 3 at. In (D), the function is not defined at x= 3, so j(x) is not continuous at x = 3.- D In order to find the equation of the tangent line, we use the equation of a line y − y 1 = m(x − x 1 ).
We are going to need to find y 1 , which we get by plugging x = into the equation for y: y = 3