Determine the following limits.

b.

Question

Determine the following limits.b.

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

Accepted Answer

Welcome back everyone. Find the limit. Limit as x approaches -1 of x + 1 divided by X to the power of 6 minus X to the power of 4th in a very given 4 answer choices A says -11/2, B 1/2, C1 and D1. So let's solve this limit. Let's begin by rewriting the limit limit as x approaches -1 of X + 1. Divided by x to the power of 6th minus x to the power of 4th, and as always we begin with direct substitution. In the numerator we're going to get -1 + 1. In the denominator, we're going to get -1s the power of 6th, minus -1s the power of 4th. So this gives us an indeterminate form which is 0 divided by 0. And if we have an indeterminate form, well, one of the ways to potentially eliminate a whole is to factor out the expression. So we're going to rewrite our limit, limit as X approaches -1. And for the denominator we're going to factor out X to the power of 4, which leaves us with X2 minus 1 is the other factor. What we notice from here is that we can factor out X2 minus 1 using the formula for factorization of the difference of squares, so our numerator remains x + 1. Our denominator becomes x to the power of 4th multiplied by x minus 1 multiplied by X + 1, because that's the factored form of X2 minus 1. What we can see is that. We can now cancel out the whole. There is a common factor X + 1. And now let's attempt direct substitution once again to see if we can get a finite number for our answer. In the numerator, we only have one remaining because we have canceled out X + 1. In the denominator we have X to the power of 4th, and since x approaches -1, we get -1 to the power of 4th multiplied by -1 minus 1. So now we end up with a finite number in the numerator we get 1. In the denominator -1 is the power of 4th is simply 1. So we get -1 minus 1, which is -2 in the denominator. So the answer is now a finite number, meaning we have successfully identified the limit, which is -12, and the answer to this problem is option A. Thank you for watching.

Determine the following limits.

b. $\lim_{x \to 2} \frac{x - 2}{x^{5} - 4 x^{3}}$

Key Concepts

Limits

Factoring Polynomials

Indeterminate Forms

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