Determine the following limits.

a.

Question

Determine the following limits.a.

\lim_{x \to 1^{+}} \frac{x - 3}{\sqrt{x^{2} - 5 x + 4}}

Accepted Answer

Welcome back everyone. Find the limit. Limit as x approaches 3 from the right of X minus 4 divided by square root of 2 X2 minus 14 X plus 24, and we're given 4 answer choices A says negative infinity, B 0 C -1, and D the limit does not exist. So let's define the limit. Limit is x approaches 3 from the right of x minus 4. Divided by square root of 2 X squared minus 14 X plus 24 and we always begin with direct substitution. Let's ignore the side. It says X approaches 3 from the right. Let's simply take X equals 3. So we get 3 minus 4 in the numerator. In the denominator we're going to get. 2 multiplied by 3 squad minus 14 multiplied by 3 + 24. Now let's try and evaluate the result in the numerator. We're going to get -1. And now for the denominator, we're going to get square root. Of Si. Now, whenever we divide a number by 0, well, this essentially gives us a limit of infinity, but to accurately tell what the limit is, well, we need to apply factorization for the denominator. In the denominator, first of all we can factor out the 2, and then in we're going to get X2 minus 7 X plus 12. Applying complete factorization, we're going to get 2 multiplied by X minus 3, multiplied by X minus 4. So let's rewrite our limit in terms of its factored form. We get limit as x approaches 3 from the right. In the numerator we get X minus 4. In the denominator, that's square root of. 2 multiplied by X minus 3 multiplied by X minus 4. And now let's analyze the behavior. So now once again applying the direct substitution, we get 3 minus 4, which is -1. And considering the square root term. We got 2 multiplied by, we're approaching 3 from the right. Which essentially means that X minus 3 approaches 0 from the right. So we're multiplying by 0 from the right, and then multiplying by 3 minus 4, which is -1. And now notice that in the square root term we get 2 multiplied by 0 from the right and then one of our factors is a negative number, and this would make all of our expression under the radical negative, right? And we have to recall that square root of X is defined as long as X is greater than or equal to 0, so our limit does not exist. Because we have a negative term. Under the radical. Therefore, the final answer to this problem corresponds to the answer choice D. The limit does not exist. Thank you for watching.

Determine the following limits.

a. $\lim_{x \to 1^{+}} \frac{x - 3}{\sqrt{x^{2} - 5 x + 4}}$

Key Concepts

Limits

Rational Functions

Indeterminate Forms

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