Determine the following limits.

b. lim t→−2^−...

Question

Determine the following limits.

b. lim t→−2^− t^3 − 5t^2 + 6t / t^4 − 4t^2

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as x approaches -7 from the left of X cubed minus 12 x2 plus 35 x divided by X to the power of 4th minus 49 x 2. We're given 4 answer choices A says 0, B1, C negative infinity, and D infinity. So let's write down the limit. And specifically we can begin factoring it out. So x approaches -7 from the left. We can factor out X in the numerator, and this gives us X multiplied by X2 minus 12 x plus 35 divided by. And now in the denominator we have a difference of two squares. We can write it as X2 minus 7 x 2. So now for the numerator, let's factor out the quadratic polynomial. We're going to get X multiplied by two factors X minus 5 and X minus 7. And now for the denominator, we have a difference of squares, which gives us X2 minus 7 X. Multiplied by X2 + 7 X. What we're going to do from here is also notice that we can factor out X or the denominators. We get limit as X approaches -7 from the left of X multiplied by X minus 5. And X minus 7 in the denominator if we factor out X and then factor out X again, we're going to get X2, in X minus 7 multiplied by X plus 7. And now we can cancel out two factors x minus 7. And also X because we have X squad in the denominator. So now let's attempt the direct substitution. We're going to get -7. -5 in the numerator, we're not including the superscript because this is a number that is not equal to 0. And now in the denominator, we have X, which is -7, multiplied by -7 from the left plus 7. This is where the superscript matters because we're going to get 0. First of all, let's simplify. So in the numerator, we're going to get -12. In the denominator, we're getting -7 multiplied by 0. And now we want to understand if it is greater. 0 or slightly less than 0. So if x approaches -7 from the left, this means that X is less than -7. So x + 7 is going to be less than 0, right, because we can add 7 to both sides. If it is less than 0, we can say we're approaching 0 from the left. Now Let's simplify it further. We're going to get -12 in the numerator. -7 multiplied by 0 gives us 0. A negative number multiplied by a negative number gives us a positive number. So eventually we're dividing -12 by 0. And a number divided by 0 gives us a limit of infinity, and now it will be negative infinity because 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.

b. lim t→−2^− t^3 − 5t^2 + 6t / t^4 − 4t^2

Key Concepts

Limits

Polynomial Functions

Factoring and Simplifying

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