Determine the following limits.
lim h→0 (h + 6)^2 + (h ...

Question

Determine the following limits.

lim h→0 (h + 6)^2 + (h + 6) − 42 / h

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with the limits. This question says to find the following limit. The limit as X approaches zero of the quantity of X plus eight in quantity squared plus the quantity of X plus eight in quantity minus 72. And that whole quantity is divided by X. We're given four possible choices as our answers. For choice A, we have 16 for choice B, we have 17 for choice C, we have 24 and for choice D we have 25. So the question asks us to find the limit as X approaches zero of the expression, which is the quantity of X plus eight in quantity squared plus the quantity of X plus eight in quantity minus 72. And that whole quantity is divided by X. Now, if we just directly plug in X equal to zero into this expression, to try to evaluate the limit in that manner, what we would get would be the expression of zero plus eight. The whole quantity is squared plus the quantity of zero plus eight in quantity minus 72. And that whole quantity is divided by zero Now, if we evaluate this expression, this is going to be equal to zero divided by zero, which is an indeterminate form. So what we're actually going to do is multiply out our numerator. So we're going to have the quantity of X plus eight in quantity squared plus quantity of X plus eight minus 72. And when I expand that out, I will get X squared plus 16 X plus 64 plus X plus eight minus 72. And when I simplify this expression, this is gonna be equal to X squared plus 17 X. So I can substitute in this result into our expression with the limit. So we'll have the limit as X approaches zero of the quantity of X squared plus 17 X. And that quantity is divided by X. So I can factor out an X in the numerator. So this is gonna be equal to the limit as X approaches zero of the quantity of X multiplying the quantity of X plus 17 in quantity divided by X, the factor of X cancels out. And I'm left with this being equal to the limit as X approaches zero of the quantity of X plus 17. Now, at this point, we can set X equal to zero to evaluate this limit. And so this is going to be equal to zero plus 17, which is equal to 17. And so that is the answer that I'm looking for. And if I look at my answer choices, this corresponds to answer choice B so I hope that this explanation has been useful and I'll see you in the next video.

Determine the following limits.
lim h→0 (h + 6)^2 + (h + 6) − 42 / h

Key Concepts

Limits

Algebraic Manipulation

L'Hôpital's Rule

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