Section 2.2 : The Limit
6. Below is the graph of f(x). For each of the given points determine the value of f(a) and limx→af(x). If any of the quantities do not exist clearly explain why.
- a=−2
- a=−1
- a=1
- a=3
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a a=−2 Show SolutionBecause there is no closed dot for x=−2 we can see that,
f(−2)does not existWe can also see that as we approach x=−2 from both sides the graph is not approaching a value from either side and so we get,
limx→−2f(x)does not existb a=−1 Show Solution
From the graph we can see that,
f(−1)=3because the closed dot is at the value of y=3.
We can also see that as we approach x=−1 from both sides the graph is approaching the same value, 1, and so we get,
limx→−1f(x)=1Always recall that the value of a limit does not actually depend upon the value of the function at the point in question. The value of a limit only depends on the values of the function around the point in question. Often the two will be different.
c a=1 Show Solution
Because there is no closed dot for x=1 we can see that,
f(1)does not existWe can also see that as we approach x=1 from both sides the graph is approaching the same value, -3, and so we get,
limx→1f(x)=−3Always recall that the value of a limit does not actually depend upon the value of the function at the point in question. The value of a limit only depends on the values of the function around the point in question. Therefore, even though the function doesn’t exist at this point the limit can still have a value.
d a=3 Show Solution
From the graph we can see that,
f(3)=4because the closed dot is at the value of y=4.
We can also see that as we approach x=3 from both sides the graph is approaching the same value, 4, and so we get,
limx→3f(x)=4