True or False. In the following exercises, justify your answer with a proof or a counterexample.
1. A function has to be continuous at if the exists.
2. You can use the quotient rule to evaluate .
3. If there is a vertical asymptote at for the function , then is undefined at the point .
4. If does not exist, then is undefined at the point .
False. A removable discontinuity is possible.
5. Using the graph, find each limit or explain why the limit does not exist.
1, starting at the open circle at (1,1).”>
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
In the following exercises, use the squeeze theorem to prove the limit.
Since , then . Since , it follows that .
18. Determine the domain such that the function is continuous over its domain.
In the following exercises, determine the value of such that the function remains continuous. Draw your resulting function to ensure it is continuous.
In the following exercises, use the precise definition of limit to prove the limit.
23. A ball is thrown into the air and the vertical position is given by . Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.
24. A particle moving along a line has a displacement according to the function , where is measured in meters and is measured in seconds. Find the average velocity over the time period .
25. From the previous exercises, estimate the instantaneous velocity at by checking the average velocity within sec.