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### Section 2.2 : The Limit

1. For the function $$\displaystyle f\left( x \right) = \frac{{8 - {x^3}}}{{{x^2} - 4}}$$ answer each of the following questions.

1. Evaluate the function at the following values of $$x$$ compute (accurate to at least 8 decimal places).
1. 2.5
2. 2.1
3. 2.01
4. 2.001
5. 2.0001
1. 1.5
2. 1.9
3. 1.99
4. 1.999
5. 1.9999
2. Use the information from (a) to estimate the value of $$\displaystyle \mathop {\lim }\limits_{x \to 2} \frac{{8 - {x^3}}}{{{x^2} - 4}}$$.

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a Evaluate the function at the following values of $$x$$ compute (accurate to at least 8 decimal places). Show Solution
1. 2.5
2. 2.1
3. 2.01
4. 2.001
5. 2.0001
1. 1.5
2. 1.9
3. 1.99
4. 1.999
5. 1.9999

Here is a table of values of the function at the given points accurate to 8 decimal places.

$$x$$ $$f(x)$$ $$x$$ $$f(x)$$
2.5 -3.38888889 1.5 -2.64285714
2.1 -3.07560976 1.9 -2.92564103
2.01 -3.00750623 1.99 -2.99250627
2.001 -3.00075006 1.999 -2.99925006
2.0001 -3.00007500 1.9999 -2.99992500

b Use the information from (a) to estimate the value of $$\displaystyle \mathop {\lim }\limits_{x \to 2} \frac{{8 - {x^3}}}{{{x^2} - 4}}$$. Show Solution

From the table of values above it looks like we can estimate that,

$\mathop {\lim }\limits_{x \to 2} \frac{{8 - {x^3}}}{{{x^2} - 4}} = - 3$