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 lim. 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,
\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 2} \right)\,\,\,{\mbox{does not exist}}}}We 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,
\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to - 2} f\left( x \right)\,\,\,{\mbox{does not exist}}}}b a = - 1 Show Solution
From the graph we can see that,
\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 1} \right) = 3}}because 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,
\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to - 1} f\left( x \right) = 1}}Always 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,
\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 1 \right)\,\,\,{\mbox{does not exist}}}}We can also see that as we approach x = 1 from both sides the graph is approaching the same value, -3, and so we get,
\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 1} f\left( x \right) = - 3}}Always 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,
\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 3 \right) = 4}}because 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,
\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 3} f\left( x \right) = 4}}