Calculus I - The Limit

Section 2.2 : The Limit

6. 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.

$a = - 2$
$a = - 1$
$a = 1$
$a = 3$

$The graph of a function on $-4 \le x \le 4$ with a vertical asymptote at $x = -2$. To the left of the asymptote the graph starts at about (-4,1.5) and increases toward positive infinity as it approaches $x = -2$. To the right of the asymptote the graph comes up from negative infinity and rises to an open dot at (-1,1). There is a closed dot at (-1,3) above the open dot at $x = -1$. From the open dot at (-1,1) the graph falls to its lowest point at an open dot at (1,-3), then rises through a closed dot at (3,4) and continues upward.$

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a $a = - 2$ 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 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,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 1} f\left( x \right) = - 3}}\]

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,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 3} f\left( x \right) = 4}}\]