Question

11-34: Evaluate the limit, if it exists.

32. \(\mathop {\lim }\limits_{x \to - 4} \left( {\frac{{\sqrt {{x^2} + 9} - 5}}{{x + 4}}} \right)\)

Short answer

The solution of the limit is \( - \frac{4}{5}\).

Step-by-step solution

The limit laws

Let \(c\) be a constant and the limits \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) and \(\mathop {\lim }\limits_{x \to a} g\left( x \right)\) exists. Then

1. \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) + g\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) + \mathop {\lim }\limits_{x \to a} g\left( x \right)\)
2. \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right) - g\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) - \mathop {\lim }\limits_{x \to a} g\left( x \right)\)
3. \(\mathop {\lim }\limits_{x \to a} \left( {cf\left( x \right)} \right) = c\mathop {\lim }\limits_{x \to a} f\left( x \right)\)
4. \(\mathop {\lim }\limits_{x \to a} \left( {f\left( x \right)g\left( x \right)} \right) = \mathop {\lim }\limits_{x \to a} f\left( x \right) \cdot \mathop {\lim }\limits_{x \to a} g\left( x \right)\)
5. \(\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}} = \frac{{\mathop {\lim }\limits_{x \to a} f\left( x \right)}}{{\mathop {\lim }\limits_{x \to a} g\left( x \right)}}\,\,{\mathop{\rm if}\nolimits} \,\,\mathop {\lim }\limits_{x \to a} g\left( x \right) \ne 0\).
6. \(\mathop {\lim }\limits_{x \to a} {\left( {f\left( x \right)} \right)^n} = {\left( {\,\mathop {\lim }\limits_{x \to a} f\left( x \right)} \right)^n}\), where \(n\) is a positive integer
7. \(\mathop {\lim }\limits_{x \to a} \sqrt[n]{{f\left( x \right)}} = \,\sqrt[n]{{\mathop {\lim }\limits_{x \to a} f\left( x \right)}}\), where \(n\) is a positive integer

(If \(n\) is even, we assume that \(\,\mathop {\lim }\limits_{x \to a} f\left( x \right) > 0\).)

1. \(\,\mathop {\lim }\limits_{x \to a} c = c\)
2. \(\,\mathop {\lim }\limits_{x \to a} x = a\)
3. \(\,\mathop {\lim }\limits_{x \to a} {x^n} = {a^n}\), where \(n\) is a positive integer
4. \(\,\mathop {\lim }\limits_{x \to a} \sqrt[n]{x} = \sqrt[n]{a}\), where \(n\) is a positive integer

(If \(n\) is even, we assume that \(a > 0\).)

 Step 2: Evaluate the limit 

Quotient law cannot be applied since the limit of the denominator is 0.

So, rationalize the numerator and evaluate the limit as shown below:

\(\begin{array}{c}\mathop {\lim }\limits_{x \to - 4} \frac{{\sqrt {{x^2} + 9} - 5}}{{x + 4}} &=& \mathop {\lim }\limits_{x \to - 4} \frac{{\left( {\sqrt {{x^2} + 9} - 5} \right) \cdot \left( {\sqrt {{x^2} + 9} + 5} \right)}}{{\left( {x + 4} \right) \cdot \left( {\sqrt {{x^2} + 9} + 5} \right)}}\\ &=& \mathop {\lim }\limits_{x \to - 4} \frac{{\left( {{x^2} + 9 - 25} \right)}}{{\left( {x + 4} \right) \cdot \left( {\sqrt {{x^2} + 9} + 5} \right)}}\\ &=& \mathop {\lim }\limits_{x \to - 4} \frac{{\left( {{x^2} - 16} \right)}}{{\left( {x + 4} \right) \cdot \left( {\sqrt {{x^2} + 9} + 5} \right)}}\\ &=& \mathop {\lim }\limits_{x \to - 4} \frac{{\left( {x + 4} \right)\left( {x - 4} \right)}}{{\left( {x + 4} \right) \cdot \left( {\sqrt {{x^2} + 9} + 5} \right)}}\end{array}\)

Solve further,

\(\begin{array}{c}\mathop {\lim }\limits_{x \to - 4} \frac{{\sqrt {{x^2} + 9} - 5}}{{x + 4}} &=& \mathop {\lim }\limits_{x \to - 4} \frac{{\left( {x - 4} \right)}}{{\left( {\sqrt {{x^2} + 9} + 5} \right)}}\\ &=& \frac{{\left( { - 4 - 4} \right)}}{{\left( {\sqrt {16 + 9} + 5} \right)}}\\ &=& \frac{{ - 8}}{{5 + 5}}\\ &=& - \frac{8}{{10}}\\ &=& - \frac{4}{5}\end{array}\)

Thus, the solution of the limit is \( - \frac{4}{5}\).