Determine the following limits.
lim x→3 x^4 − 81 / x − 3...

Question

Determine the following limits.

lim x→3 x^4 − 81 / x − 3

Accepted Answer

Hello there. So today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the limit of F of X equals X to the power of four minus 16 divided by X minus two as X approaches two. Awesome. So it appears for this particular prong, we're trying to figure out what the limit of this F of X function is as X approaches two. So now that we know that we're trying to figure out what the limit is, let's read off our multiple choice answers to see what our final answer might be. A is 16. B is negative four C is 32 and D is eight. OK? So first off what we need to do in order to help us solve this problem is we need to identify the target variable and the limit expression. So ultimately, we're trying to figure out the limit as X approaches two of X to the power of four minus 16, all divided by X minus two. OK. So the first way we could try solving a problem like this is we can perform direct substitution. So let's do that first to see if we could use this method to solve our limit. So we need to take the limit as X approaches two of X to the power of four minus 16, all divided by X minus two. And this is equal to, when we plug in the value for two, for X, we'll take two to the power of four minus 16, divided by two minus two. So when we perform that, we will find that it's equal to zero, divided by zero because two to the power of four is 16 and 16 minus 16 is zero and two minus two is zero. But it's important to note and recall that since zero divided by zero is an indeterminate form, this particular limit cannot be found by direct substitution. So we cannot use this method. So instead, our next way of solving is we need to use the factor method. So we need to factor the numerator instead. So, so as we should recall, we need to factor the numerator instead. So when we factor the numina numerator, which is the upper part of the division, the top part of the division bar, we need to notice that X to the power of four minus 16 can be factored as X squared minus four, multiplied by X squared plus four. And further X squared minus four can be factored as X minus two multiplied by X plus two. So therefore, we could go ahead and write that X to the power of four minus 16 divided by X minus two. And let's put these values in parentheses will be equal to X minus two, multiplied by X plus two, multiplied by X squared. So multiplied by X squared plus four divided by X minus two. Fantastic. So we can simplify this expression. So we need to cancel out the common factor of X minus two. So this will give us the following expression when we simplify X plus two, multiplied by X to the power of two plus four Fantastic. So we can now evaluate the limit. So now we can plug in and let's make a note of this in red, we can plug in X equals two into our simplified expression. So when we do that, we will get two plus two and then we need to multiply this by two to the power of two plus four. And when we do this, when we do this, when we plug it into our calculator, we will find that this is equal to 32 and that's it we've officially solved for this problem. Hooray. So therefore, our limit of uh so our limit of X approaches two of X to the power four minus 16 divided by X minus two. This is all equal to 32. And this corresponds with multiple choice answer letter C which once again, is 32. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Determine the following limits.
lim x→3 x^4 − 81 / x − 3

Key Concepts

Limits

Factoring Polynomials

Indeterminate Forms

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