Help with limits

[izzlebop]

I can't seem to get the equation to a point where the denominator doesn't equal to zero when the 4 is sub'd in.?

lim (4 - x) / ( 5 - sqrt(x^2 +9) )
x ->4

Alternatively, is there another way to approach this other than subbing the 4?

 

[pappus]

I can't seem to get the equation to a point where the denominator doesn't equal to zero when the 4 is sub'd in.?

lim (4 - x) / ( 5 - sqrt(x^2 +9) )
x ->4

Alternatively, is there another way to approach this other than subbing the 4?

\(\displaystyle \lim_{x \to 4}\left(\frac{4-x}{5-\sqrt{x^2+9}} \right)=\lim_{x \to 4}\left(\frac{(4-x)(5+\sqrt{x^2+9})}{(5-\sqrt{x^2+9})(5+\sqrt{x^2+9})} \right)=\lim_{x \to 4}\left(\frac{(4-x)(5+\sqrt{x^2+9})}{(4-x)(4+x} \right)\)

Continue!

 

[srmichael]

I can't seem to get the equation to a point where the denominator doesn't equal to zero when the 4 is sub'd in.?

lim (4 - x) / ( 5 - sqrt(x^2 +9) )
x ->4

Alternatively, is there another way to approach this other than subbing the 4?

Izzlebop, as you have found out from this problem, you can't always "substitute in" the valule that x is approaching into the limit to evaluate. In fact, most of the time you can't otherwise these problems would all be too easy . Instead, you usually have to manipulate the limit into a form where you can get to the point where you can "substitute in" the value that x is approaching.

In this case, as Pappus displayed, you need to multiply the numerator and the denominator by the conjugate of the denominator. When simplifying, you will see that there will be a cancellation that will then allow you to "substitute in" the value that x is approaching.

Make sense?