\f(3) Find the following limit, if it exists Jim (vx2 + x - x) ( A) O (B) 1/ (C) 1/2 (D) Does not exist (E) None of the above +(4) Fill-in-the-blank: Use an & - 8 method to prove the following limit lim (5x - 3) = 2 Note: Substituting x = 1 into the expression to verify the limit, is not a proof using the a - 8 method Proof: (Fill-in-the-blank) (1) Find a value for 8: (i) (2 pts) We need to show that for every & > 0, there exists a 8 > 0 such that: (ii) (2 pts) In particular, we need to show that for every => 0, there exists a : > 0 such that: (iii) (6 pts) And since, every & > 0, there exists a 8 > 0, such that: , we need to show for (iv) (1 pt) Suggesting a good candidate for & would be: (1/) Confirm that the value : works: v) (4 pts) To confirm the & candidate is valid for a given & > 0, choose 8 = and assume (vi) (2 pts)Then consider: (vii) (2 pts) Therefore, if , then (vili) (1 pt) Which proves by definition our limit statement: +(5)_ Find the equation of the tangent line to the curve y = vx at x = c. Write your answer in slope-intercept form, y = mx + b. (Note: You must use the definition of the slope of the tangent, no "short-cuts" yet.) (A) y = 27 (B )y = zax+c (C)y= cvx+ vc (D)y= zx+2 (E) None of the above +(2)_ Use the graphs of F and G in Figure 1 and the Limit Principles to find the following limits provided they exist: (i) lim [2 F(x) - 3 G(x)] (ii) lim [F(x)]3 + lim [G(x)]2 (ui) lim [F(x)G(x)] +3 (iv) limc(x) F(x) Figure 1 F(x) G(x (A) (1)5 (ii) 9 (iti) 1 (iv) 3 (B) (1)3 (ii)-4 (iii) 4 (iv) 1 (C) (1)0 ii) 9 (iii) does not exist (iv) 3 (D) (1)5 (ii) 9 (iii) does not exist (iv) 3 (E) none of the above