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Section 2.5 : Computing Limits

7. Evaluate \(\displaystyle \mathop {\lim }\limits_{z \to 4} \frac{{\sqrt z - 2}}{{z - 4}}\), if it exists.

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There is not really a lot to this problem. Simply recall the basic ideas for computing limits that we looked at in this section. In this case we see that if we plug in the value we get 0/0. Recall that this DOES NOT mean that the limit doesn’t exist. We’ll need to do some more work before we make that conclusion. If you’re really good at factoring you can factor this and simplify. Another method that can be used however is to rationalize the numerator, so let’s do that for this problem.

\[\mathop {\lim }\limits_{z \to 4} \frac{{\sqrt z - 2}}{{z - 4}} = \mathop {\lim }\limits_{z \to 4} \frac{{\left( {\sqrt z - 2} \right)}}{{\left( {z - 4} \right)}}\frac{{\left( {\sqrt z + 2} \right)}}{{\left( {\sqrt z + 2} \right)}} = \mathop {\lim }\limits_{z \to 4} \frac{{z - 4}}{{\left( {z - 4} \right)\left( {\sqrt z + 2} \right)}} = \mathop {\lim }\limits_{z \to 4} \frac{1}{{\sqrt z + 2}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{4}}}\]