Limits Problem

You should show us what have you tried so far.
Try to factorize the numerator. If you get stuck, write back.
 
[MATH]\lim_{x \to -5} \dfrac{(x+5)(x-3)}{|x+5|}[/MATH]
you need to look at both one sided limits ...

[MATH]\lim_{x \to -5^-} \dfrac{(x+5)(x-3)}{|x+5|}[/MATH] and [MATH]\lim_{x \to -5^+} \dfrac{(x+5)(x-3)}{|x+5|}[/MATH]
 
If x=-5 makes both the numerator and denominator equal to 0, then one factor for the numerator and denominator will be x+5.

Also check both sided limits and cancel carefully.

What does |x+5| equal without the absolute value bars?
 
I factored the numerator and plugged in -5 and the denominator ends up with 0
Yes, it does. Now what about the numerator? (5)2+2(5)15=251015=0\displaystyle (-5)^2+ 2(-5)- 15= 25- 10- 15= 0 also so both numerator and denominator are 0!
View attachment 21600

Does the limit not exist for this?
[/QUOTE]
Think about limx0xx\displaystyle \lim_{x\to 0} \frac{x}{x}. The denominator goes to 0 (as does the numerator) but, for x not 0, this is precisely 1. It's limit exists and is 1.

You say you factored the numerator and, I presume, you got (x+ 5)(x- 3). x3+2x15x+5=(x+5)(x3)x+5\displaystyle \frac{x^3+ 2x- 15}{|x+ 5|}= \frac{(x+ 5)(x- 3)}{|x+ 5|}.

For x> -5, x+5=x+5\displaystyle |x+ 5|= x+ 5 so this is (x+5)(x3)x+5=x3\displaystyle \frac{(x+ 5)(x- 3)}{x+ 5}= x- 3. The limit of that, for x going to -5, is -5- 3= -8.

For x< -5. x+5=(x+5)\displaystyle |x+ 5|= -(x+ 5) so this is (x+5)(x3)(x+5)=(x3)\displaystyle \frac{(x+ 5)(x-3)}{-(x+ 5)}= -(x- 3). The limit of that, for x going to -5, is (58)=8\displaystyle -(-5- 8)= 8.

The limit does not exist but because the two one-sided limits are not the same, not because the denominator is 0. If the problem had been limx5x3+2x15x+5\displaystyle \lim_{x\to -5}\frac{x^3+ 2x- 15}{x+ 5}, without the absolute value, the limit would be -8 even though the denominator still goes to 0.
 
Yes, it does. Now what about the numerator? (5)2+2(5)15=251015=0\displaystyle (-5)^2+ 2(-5)- 15= 25- 10- 15= 0 also so both numerator and denominator are 0!
View attachment 21600

Does the limit not exist for this?
Think about limx0xx\displaystyle \lim_{x\to 0} \frac{x}{x}. The denominator goes to 0 (as does the numerator) but, for x not 0, this is precisely 1. It's limit exists and is 1.

You say you factored the numerator and, I presume, you got (x+ 5)(x- 3). x3+2x15x+5=(x+5)(x3)x+5\displaystyle \frac{x^3+ 2x- 15}{|x+ 5|}= \frac{(x+ 5)(x- 3)}{|x+ 5|}.

For x> -5, x+5=x+5\displaystyle |x+ 5|= x+ 5 so this is (x+5)(x3)x+5=x3\displaystyle \frac{(x+ 5)(x- 3)}{x+ 5}= x- 3.[/quote]
I think one should specifically say that "(x+5)(x3)x+5=x3\displaystyle \frac{(x+ 5)(x- 3)}{x+ 5}= x- 3" is only true for x not equal to 5. Of course, since the "limit as x goes to 5" does not depend upon the value at x= 5, the limit is still correct.

 
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