Solution

(a)    4: The function is heading toward a height of 4 when x approaches -5 from the right. Even though the function never actually reaches that height (hence the open circle), that is still where it is headed, which is the most important characteristic of a limit.

(b)    0: The function approaches a height of 0 as you approach -5 from the left. In fact, the function contains the point (-5,0), so the function actually arrives at the height to which it was headed (but our answer wouldn't have changed even if the circle was open instead of closed at that point on the graph).

(c)    Does not exist: In order for a general (not a right- or left-hand) limit to exist, the right- and left-hand limits for that value of x must be equal. Our answers for (a) and (b) represent those limits. Because those answers are not equal, no limit can exist. If they were equal, then the general limit would be that equivalent value.

(d)    1: The graph appears to approach a height of 1 when you approach x = -2 from the left or the right.

(e)    Does not exist (or infinity): The graph increases without bound as you approach 0, so there is no limit. Note: Saying that the limit is infinity is a descriptive way of saying that the limit does not exist, as it also gives the reason why. Infinity is not a valid limit.

(f)    0: As x gets infinitely large, the graph gets closer and closet to the x-axis, which has a height of 0, so the answer is 0.

Thanks to Peter Chan for catching some typos I'd made in this problem!

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AP Calculus II (BC): Continuity

A function g(x) is defined as follows

Find the values of b and c which make g continuous.

Difficulty: 

Solution

The graph of g is one seriously weird piece of garbage. It is linear until x equals -2 and then turns into a parabola until x = 4. After that, the graph is linear again.

We just have to design the parabola so that it connects those two segments together. To do that, we'll need to figure out what those two points are. Plug x = -2 into the first linear equation and x = 4 into the second. These are the correct left-hand limits and right-hand limits respectively, which is why we don't have to worry about y = 2x - 1 being defined at x = 4.

Therefore, we know the parabola must contain the points (-2,13/2) and (4,7) or it will not meet up with the line segments correctly. We can plug each of these (x,y) coordinates into the parabola separately to get a system of equations:

You can solve this system just like any system you did in beginning Algebra (although the answer will be a little stranger than beginning algebra problems, I grant you). The quickest way is to graph both lines on the calculator and calculate the intersection point. The final answer is , and you can verify this graphically by plugging the values into the parabola. The pieces should match up exactly, creating a continuous function.

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