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So I am trying to evaluate

$\lim\limits_{x\to -4} |x+4|$

My problem solving method is as follows:
1) See if there is anything I can factor
2) If there are square roots, then conjugate
3) Determine if it does not exist (confused about when to do which)


I have no idea what to do with this problem. I can't conjugate or factor it. And I'm not sure what the absolute value means in the context of a limit.

Any advice guys? Thanks!

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Well. If $x$ is close to $-4$ then $|x+4|$ is close to ... –  David Mitra Feb 2 at 22:41
    
Just substitute $x=-4$ into $x+4$, and take the absolute value. That's all it actually means (it's a whole different case if you're using L'Hospital's rule, though). –  Sanath Devalapurkar Feb 2 at 22:41
    
@DavidMitra: I think you meant $|x+4|$, am I right? –  Sanath Devalapurkar Feb 2 at 22:42
    
@SanathDevalapurkar Yes. It's right now. –  David Mitra Feb 2 at 22:43
    
If you've discussed the concept of continuity in your course, you will find that when a function $ \ f(x) \ $ is continuous at a point $ \ x \ = \ a \ , $ you may calculate the limit as $ \ x \ \rightarrow \ a \ $ by "direct substitution" , since $ \lim_{x \rightarrow a} \ f(x) \ = \ f(a) \ $ when continuity pertains. The absolute value function, even when shifted horizontally as in this problem, is continuous for all real numbers. –  RecklessReckoner Feb 2 at 22:49
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2 Answers

up vote 2 down vote accepted

There's a first step that you're missing in your method.

Hint: how would you find $$ \lim_{x \to -4} (x+4) $$ and why is this situation similar?

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Is that first step, plugging it in? EDIT: Wait. I think I have it. I think the answer is zero. I am used to not being able to use zero because I deal with rational expressions, but in this case zero has no limitations. The first step you are referring to I think it to identify limitations, in this case there are none. So the limit is zero –  taz henery duck0 Feb 2 at 22:59
    
Indeed, the first step in taking the limit (of any function that's continuous around the point in question) always to "plug it in". If your limit produces something that is not an indeterminate form (such as $\frac 00$, $\frac{\infty}{\infty}$, or $1^{\infty}$), then the limit is simply what comes out when you plug in. If you get an indeterminate form though, you'll have to try another way (i.e. by factoring, multiplying by the conjugate, etc.). –  Omnomnomnom Feb 2 at 23:06
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One of the most useful rules is that if $f$ is continuous at $x=a$, then $\lim_{x \rightarrow a} f(x)= f(a)$. Of course, it's often the case that you need to evaluate the limit precisely because you are being asked to prove that the function is continuous at that point, so you can't use this fact then (circular reasoning).

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