The graphs of f and g are given. Use them to evaluate each limit, if it exists. (If an answer does not exist, enter DNE.)The x y coordinate plane is given. There is one curve and one point on the graph. The point occurs at (−1, 3). The curve enters the window at (−2, −1), goes up and right, passes through the open point (−1, 1), changes direction at (0, 2), goes down and right, and passes through the open point (2, −1). At this point, the curve turns sharply forming a cusp, goes up and right, and exits the window at (4, 2). The x y coordinate plane is given. There are two curves on the graph. The first curve enters the window at (−2, 0), goes up and right, and ends at the closed point (0, 3). The second curve begins at the open point (0, 1), goes up and right, and passes through (2, 2). At this point, the curve turns sharply forming a cusp, goes down and right traveling in a straight line, and exits the window at (4, −2).(a)lim x→2 [f(x) + g(x)](b)lim x→0 [f(x) − g(x)](c)lim x→−1 [f(x)g(x)](d)lim x→3 f(x)g(x)(e)lim x→2 [x2f(x)](f)f(−1) + lim x→−1 g(x)

Use the graph of g to find the value of each expression. (If an answer does not exist, enter DNE.)The x y coordinate plane is given. The curve enters the window in the second quadrant, goes down and right to the closed point (0, −1). The curve starts again at the open point (0, −2), goes up and right to the open point (2, 2). There is a closed point at (2, 1). The curve starts again at the open point (2, 0), goes up and right, passes through the point (4, 3), and exits the window in the first quadrant.(a)lim x → 0− g(x)(b)lim x → 0+ g(x)(c)lim x → 0 g(x)(d)lim x → 2− g(x)(e)lim x → 2+ g(x)(f)lim x → 2 g(x)(g)g(2)(h)lim x → 4 g(x)

For the function A whose graph is shown, state the following. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)The x y-coordinate plane is given. The function enters the window in the second quadrant, goes up and right becoming more steep, exits just to the left of x = −3 in the second quadrant nearly vertical, reenters just to the right of x = −3 in the second quadrant nearly vertical, goes down and right becoming less steep, crosses the x-axis at x = −2, goes down and right becoming more steep, exits the window just to the left of x = −1 in the third quadrant nearly vertical, reenters just to the right of x = −1 in the third quadrant nearly vertical, goes up and right becoming less steep, crosses the y-axis at approximately y = −0.6, changes direction at the approximate point (0.5, −0.5) goes down and right becoming more steep, exits the window just to the left of x = 2 in the fourth quadrant nearly vertical, reenters just to the right of x = 2 in the first quadrant nearly vertical, goes down and right becoming less steep, crosses the x-axis at x = 3, changes direction at the approximate point (4.5, −1.5), goes up and right becoming more steep, crosses the x-axis at approximately x = 6.5, and exits the window in the first quadrant.(a)lim x → −3 A(x) (b)lim x → 2− A(x) (c)lim x → 2+ A(x) (d)lim x → −1 A(x) (e)The equations of the vertical asymptotes. (Enter your answers as a comma-separated list.)

The graph of a function g is shown. The x y-coordinate plane is given. The curve begins at the point (−2, 0), goes up and right, passes through the point (−1.5, 1), goes up and right, changes direction at the point (−1, 1.5), goes down and right, passes through the point (−0.5, 1), goes down and right, passes through the origin, goes down and right, passes through the point (0.5, −1), goes down and right, changes direction at the point (1, −1.5), goes up and right, passes through the point (1.5, −0.5), goes up and right, changes direction at the point (2, 0.5), goes down and right, crosses the x-axis at x = 2.5, goes down and right, changes direction at the point (3, −1), goes up and right, passes through the point (3.5, −0.5), goes up and right, and ends at the point (4, 0.5). Estimate 4 −2 g(x) dx with six subintervals using the following. (a) right endpoints 0 Correct: Your answer is correct. (b) left endpoints -0.5 Correct: Your answer is correct. (c) midpoints

For the function f whose graph is shown, state the following. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)FigureThe x y coordinate plane is given. Refer to the adjacent description for more details.Description(a)lim x → −7 f(x)(0,2) (b)lim x → −3 f(x)DNE (c)lim x → 0 f(x)2 (d)lim x → 6− f(x)∞ (e)lim x → 6+ f(x)−∞ (f)the equations of the vertical asymptotes (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)x = (−3,6)

The graph of a function g is shown.The x y-coordinate plane is given. A function labeled y = g(x) with 4 parts is graphed.The first part is a curve, enters the window in the second quadrant, goes up and right becoming less steep, crosses the y-axis at approximately y = 2.5, and ends at the open point (2, 3).The second part is a curve begins again at the open point (2, 1), goes up and right becoming less steep, and ends at the open point (5, 2).The third part is the closed approximate point (5, 1.2).The fourth part is a curve, begins at the open point (5, 2) goes down and right becoming more steep, and exits the window in the first quadrant.Use it to state the values (if they exist) of the following:(a)  lim x → 2− g(x)(b)  lim x → 2+ g(x)(c)  lim x → 2 g(x)(d)  lim x → 5− g(x)(e)  lim x → 5+ g(x)(f)  lim x → 5 g(x)SolutionLooking at the graph we see that the values of g(x) approach as x approaches 2 from the left, but they approach as x approaches 2 from the right.Therefore (a) lim x → 2− g(x) =     and    (b) lim x → 2+ g(x) = .Since the left and right limits are different, we conclude that (c) the limit as x approaches 2 of g(x) does not exist.The graph also shows that (d) lim x → 5− g(x) =     and    (e) lim x → 5+ g(x) = .This time, the left and right limits are the same and so, by this theorem, we have (f) lim x → 5 g(x) = Despite this fact, notice that g(5) ≠ 2.