Find the following limits or state that they do not exist. Ass...

Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→9 √x − 3 / x − 9

Accepted Answer

Welcome back, everyone find the limit below or state that it does not exist limit of square root of X minus six divided by X minus 36. When X approaches 36 there are four answer choices. A 16 B 1/12 C 1/18 and D, the limit does not exist. So first of all, we want to apply direct substitution limit when X approaches 36 of square root of X minus six divided by X minus 36. When we apply the direct substitution, we end up with square root of 36 minus six, divide by 36 minus 36. This gives us zero divided by zero, which is an indeterminate form. So we want to somehow eliminate this in the terminate form. And we can do that using factorization. Let's carefully look at the denominator. And let's recall that whenever we have A squared minus B squared, we can factor out the difference of squares as A minus B multiplied by A plus B. In this case, we can rewrite X is square root of X squared and 36 S six squared, right. So square root of X squared minus 36 or basically minus six squared can be factored out as squared of X minus six, multiplied by a score of X plus six. And this is really useful because we have the same numerator as one of our factors. So we can rewrite this limit as limit when X approaches 36 of squared of X minus six, divided by square root of X minus six multiplied by squared of X plus six. And this allows us to cancel out one of the factors squared of X minus six. So the limit becomes limit when X approaches 36 of one divided by square root of X plus six. And now let's apply direct substitution once again. So A approaches 36 meaning we get one divided by square root of 36 plus six, which is one divided by six plus six. That's 12. This is a finite number, meaning we have successfully evaluated the limit and the answer is B 1/12. Thank you for watching.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→9 √x − 3 / x − 9

Key Concepts

Limits

Indeterminate Forms

Rationalizing the Numerator

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