Finding Limits
In Exercises 9–24, find ...

Question

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim x→a (x² ― a²)/(x⁴ ― a⁴)

Accepted Answer

Welcome back everyone. Determine the limit. Limit as x approaches 2C of X minus 2C divided by 4 multiplied by X cubed minus 8 C cubed. We're given 4 answer choices A says 1 divided by 12c 2, B 1 divided by 3 C2, C 1 divided by 4C2 and D 1 divided by 48C2. So let's rewrite the limit. The X approaches to C of X minus 2C divided by or multiplied by X cubed minus 8 C cubed. As always, we begin with direct substitution. Let's suppose that X is equal to 2 C, so we get 2c minus 2C in the numerator and in the denominator that's 4 multiplied by 2c cubed, which is 8 c cubed minus 8C cubed. So this gives us an indeterminate form which is 0 divided by 0. For this in determinant form, if we have a rational function, one of the ways to solve for the limit is to factor out. So we're going to factor out the denominator because it is a difference of cubes. We have X cubed minus 2c cubed. That's the same as 8 C cubed, right? And now let's recall the factorization formula. If we have a cubed minus b cubed, this can be factored as a minus B multiplied by a squared. Plus AB + B squad. Our A is X and our B is 2C. So we get X minus 2C. Multiplied by Aquad, so X2 plus A B, so X multiplied by 2 C, we get 2 C X. Plus B2, so we square 2 C which gives us 4C2. And this means that our limit becomes limit as x approaches to C of X. Minus 2 C, all of that divided by 4, and then we can write the factor form X minus 2C multiplied by X2 + 2 C X + 4C squad. Notice that now we can cancel out the whole x minus 2 C. Which will possibly help us. Evaluate The value of the limit. So we have 1 in the numerator, we have 4 in the denominator, and now let's apply the rank substitution. So 2c 2d gives us. Or C2 + 2 CX, so 2C multiplied by 2C gives us 4 C squad, and we're adding another 4C squad. So now what we're going to do is simply take 1 in the numerator and we should notice that we have 4 multiplied by 12c 2d. So that's 48c 2d. Since we're also dividing by a dividing by 48 C2, we have to make sure that C in this context is not equal to 0, right? And that's it. We have evaluated the final limit which corresponds to the answer choice D. Thank you for watching.

Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.

lim x→a (x² ― a²)/(x⁴ ― a⁴)

Key Concepts

Limits

Rational Functions

Factoring and Simplifying

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