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### Section 2.2 : The Limit

4. Below is the graph of $$f\left( x \right)$$. For each of the given points determine the value of $$f\left( a \right)$$ and $$\mathop {\lim }\limits_{x \to a} f\left( x \right)$$. If any of the quantities do not exist clearly explain why.

1. $$a = - 3$$
2. $$a = - 1$$
3. $$a = 2$$
4. $$a = 4$$ Show All Solutions Hide All Solutions

a $$a = - 3$$ 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 different values (4 from the left and -2 from the right). Because of this we get,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to - 3} f\left( x \right)\,\,\,{\mbox{does not exist}}}}$

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.

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 $$a = 2$$ Show Solution

Because 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 approaching the same value, 1, and so we get,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 2} f\left( x \right) = 1}}$

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. Therefore, even though the function doesn’t exist at this point the limit can still have a value.

d $$a = 4$$ Show Solution

From the graph we can see that,

$\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 4 \right) = 5}}$

because the closed dot is at the value of $$y = 5$$.

We can also see that as we approach $$x = 4$$ from both sides the graph is approaching the same value, 5, and so we get,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 4} f\left( x \right) = 5}}$