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opus

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## Homework Statement

Find the following limit:

$$\lim_{x \rightarrow 10}\frac{x-10}{4-\sqrt{x+6}}$$

## Homework Equations

## The Attempt at a Solution

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Please see attached work. I have a few questions (other than if my solution is correct or not).

First, is at step (ii)(C):

What makes me uneasy about this step is the fact that I multiplied the fraction by ##\frac{-1}{-1}##, and only multiplied on the bottom initially. What this did was made the denominator term equal to one of the numerator terms so they could be cancelled. Is this legitimate? I feel like when I do this, I have to multiply both out at the same time. My recitation leader said that it was legitimate as the (-1) in the numerator is still there, we're just saving it to be multiplied through later. But it seems to me like this shouldn't work because the top and bottom should switch

*at the the same*, so we shouldn't be able to get them to the same thing to cancel.

My second question is the concept behind the entire problem:

For the given function, if we evaluate at x=10, we get ##\frac{0}{0}## which I was told is not good. So to get around this, we can change the form of the function to something equivalent, which was the bulk of this entire problem. So after the given function, we found something equivalent to it, ##y=-4-\sqrt{x+6}##, which is identical to the given function except this one has a hole at x=10. I'm having a difficult understanding what we're doing here and why. I know that we want to find the value that y approaches as the x closes in on x=10 from both sides, but the ##\frac{0}{0}## thing causing us to find an equivalent function with a hole at the value we're taking the limit at is confusing me a bit.

This just seems like Algebra to me, not Calculus as we're just taking a function and evaluating it at some x. So I guess I don't see the picture of what the problem is trying to get across to me.

Thank you for your time.