Chapter 11.2, Problem 87E

To determine

To write: a short paragraph explaining your reasoning.

Answer to Problem 87E

The rational fraction was explained.

Explanation of Solution

Given:

Rational Fraction

  $p\left(x\right)/q\left(x\right)$

Calculation:

Part A:

We know that the limit of this function as $x\to c$ is 0. If a rational function has a non-zero denominator for $x=c$ then the limit of that function can be found by direct substitution. So in this case, we know that since the denominator of this function is nonzero, then the limit is equal to what is obtained by direct substitution: $\frac{0}{1}=0$ .

Part B:

Again, by the same reasoning as part a, we can say that the limit of this function as $x\to c$ is 1, the value produced by direct substitution, since the rational function has a non-zero denominator for $x=c$ .

Part C:

We can say that the limit of this function as $x\to c$ is either $\infty$ or does not exist. We can do no manipulation to try obtain a limit, since direct substitution does not produce the indeterminate form.

Part D:

We can come to no conclusion about the limit of this function. Because direct substitution produces the indeterminate form, it is possible that we may be able to obtain a limit algebraically by employing the "Dividing Out" technique, or rationalizing technique. Or, the limit may not exist.

Conclusion:

Therefore, the rational fraction was explained.