Find the limits in Exercises 53–58. Write ∞ or −∞ where approp...

Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 4x) as

b. x→−2⁺

Accepted Answer

Welcome back, everyone. Find the limit. Limit as X approaches -3 from the right of X2 minus X minus 6 divided by X cubed minus 9 X. A says -1, B 0 C negative infinity, and D infinity. So let's go for this limit. Limit as X approaches -3 from the right. If we analyze the given rational function, we will notice that our denominator becomes equal to 0 if X is equal to -3. So division by zero is undefined. Essentially, we want to sulfur the limit without the direct substitution. And what we have to notice is that we have two polynomials in the numerator and the denominators, the 2nd degree and the 3rd degree polynomials. So what we're going to do is simply perform factorization to eliminate any potential holes. In the numerator, we have a quadratic polynomial x2 minus x minus 6, which can be factored out as X minus 3 multiplied by X + 2. For the denominator, we can take out X, which leaves us with X2 minus 9, and we know that X2 minus 9 is a difference of two squares, right? X2 minus 32. So we can factor out further. By applying the difference of squares factorization, we have x minus 3 multiplied by x + 2 in the numerator and the denominator that's X multiplied by X minus 3 multiplied by X + 3, and this is where we can cancel out the whole at x equals. 3 right, because we have a common factor. And now let's write down the resultant limit as X approaches -3 from the right of X + 2 divided by X multiplied by X + 3. And now we can show an attempted substitution. Which essentially gives us -3 + 2. Divided by -3 multiplied by -3 + 3. We can see that we're still dividing by 0, but we are not really trying to divide it, right? We just want to visualize what the expression becomes. So our numerator, -3 + 2 becomes -1. It is -1. And now considering our denominator, we have -3 multiplied by -3 from the right plus 3. Now, if X approaches -3 from the right, this means that X is greater than -3. If it's greater than -3, then we can say that -3. Plus 3. is going to be greater than 0, right? Because we have -3 from the right. This number is greater than -3, and we are adding 3. So overall, the sum is greater than 0. So we can sell that in the denominator, -3 from the right plus 3 is 0 from the right. This is 0 from the right, which essentially means that this is a negligibly small positive number, which is the most important thing. Let's recall that whenever we divide a non-zero number. By 0 when it comes to limits, we get some sort of infinity, right? And in this context, if we're dividing a negative number by a product between a negative number and a positive number, which gives us a negative number overall, we get a ratio of 2 negative numbers, which is a positive number, meaning the correct answer to this problem corresponds to the answer choice the positive infinity. Thank you for watching.

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 4x) as

b. x→−2⁺

Key Concepts

Limits

Rational Functions

One-Sided Limits

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