Solving One-Sided Limit: lim x→4⁻√x-2/x-4

[JennyInTheSky]

Homework Statement

lim$$_{x\rightarrow4^{-}}$$ $$\frac{\sqrt{x}-2}{x-4}$$

Homework Equations

Typical methods used in solving one-sided limit.

The Attempt at a Solution

I plug in something a little bit smaller than four, like 3.999999 into x, and I get $$\frac{something a little less than zero}{something a little less than zero}$$ to equal 1. But the answer is 1/4. How is this so?

 

[sutupidmath]

Homework Statement

lim$$_{x\rightarrow4^{-}}$$ $$\frac{\sqrt{x}-2}{x-4}$$

Homework Equations

Typical methods used in solving one-sided limit.

The Attempt at a Solution

I plug in something a little bit smaller than four, like 3.999999 into x, and I get $$\frac{something a little less than zero}{something a little less than zero}$$ to equal 1. But the answer is 1/4. How is this so?

But see that the function at x=4 is not defined, right? so if you just keep plugging nr. you will get zero over zero, what i would do here, is multiply top and bottom by the conjugate of your numerator, this way things will clean up nicely.

Then, you can keep plugging nrs whicha re very very close to 4 from the left, in order to determine the sign. What i alway do is i introduce an e>0, e-->0 (e stands for epsilon) and so i let then x-->4-e, but that's just me.

Because sometimes it is not sufficient to take numbers close to, say 4 in this case. etc

 

[Focus]

But see that the function at x=4 is not defined, right? so if you just keep plugging nr. you will get zero over zero, what i would do here, is multiply top and bottom by the conjugate of your numerator, this way things will clean up nicely.

Then, you can keep plugging nrs whicha re very very close to 4 from the left, in order to determine the sign. What i alway do is i introduce an e>0, e-->0 (e stands for epsilon) and so i let then x-->4-e, but that's just me.

Because sometimes it is not sufficient to take numbers close to, say 4 in this case. etc

You could just factorise the bottom $$x-4=(\sqrt{x}-2)(\sqrt{x}+2)$$

 

[sutupidmath]

You could just factorise the bottom $$x-4=(\sqrt{x}-2)(\sqrt{x}+2)$$

Either way!...lol... whatever works for you...