Determine the following limits.

Question

\lim_{z \to 4} \frac{z - 5}{{(z^{2} - 10 z + 24)}^{2}}

Accepted Answer

Welcome back, everyone. Find the limit. Limit as k approaches 7 of 2k minus 17 divided by k2 minus 13K plus 422. We're given 4 answer choices A says infinity, B 0, C negative infinity, and D does not exist. So let's rewrite the limit. We're given the limit as K approaches 7 of 2k minus 17 divided by. K2 minus 13K plus 42 2. As always, let's begin solving this limit by direct substitution. In the numerator, we get 2 multiplied by 7 minus 17, and in the denominator we're going to get 7 squad minus 13 multiplied by 7 + 42. All of this squared. Now let's simplify. In the numerator, we're going to get 2 multiplied by 7 minus 17, which gives us -3. In the denominator, 72 minus 13 multiplied by 7 + 42 gives us 0. And now we are squaring that 0, right? Now, first of all, we want to make sure that whenever we get a form where we are dividing a number by 0, well, we simply want to make sure that a limit exists. Let's understand that we want to check 2 limits when K approaches 7 from the left and when K approaches 7 from the right. If we are approaching K from one side and then the other, well, we are going to get two different zeros. One is going to be 0 from the left and the other one is going to be 0 from the right. But in this case, since we are squaring 0, whether we square 0 from the left or from the right, it doesn't really matter, right, because this is still going to be 0 from the right. Whenever we square a negative number, it becomes a positive number. So the limit will exist. Since If we approach, if we approach 7 from the left or from the right, this is going to be 0 from the right in each case. And we can simplify the form as -3 divided by 0 from the right. Whenever we divide a number by 0, this gives us a limit of infinity, and in this case this is going to be negative infinity since we're dividing a negative number by a positive number. So the correct answer to this problem corresponds to the answer choice C negative infinity. Thank you for watching.

Determine the following limits.

$\lim_{z \to 4} \frac{z - 5}{{(z^{2} - 10 z + 24)}^{2}}$

Key Concepts

Limits

Factoring Polynomials

Indeterminate Forms

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