Limits Help (Stewart Calculus, 2.3)

[bhaktir]

10. (a) What is wrong with the following equation?
(x^2 + x - 6)/(x-2) = x +3

^ That's the first part, and I'm kinda stumped. Is it incorrect because both sides are equivalent?

 

[galactus]

If both side are equivalent, then how is it incorrect?.

Factor the numerator in the left and cancel the x+3. You can then see the right hand side is all that remains.

Thus, they are equivalent. I reckon you already knew that.

One small issue. What about if x=2?.

 

[bhaktir]

Thanks!

And that's actually the second part.

lim (x-->2) on each side, and you're asked to explain why it's correct.

 

[mmm4444bot]

I'm wondering what's wrong with your subject line. I see no limit statement.

The posted equation is an "identity". This means that the equation is true for all values of x for which both sides are defined.

Are you sure that you properly typed the entire exercise? The question about incorrectness does not make sense, to me.

 

[mmm4444bot]

you're asked to explain why it's correct.

Your pronoun "it" above is unreferenced. What are you trying to say?

 

[bhaktir]

^ Sorry, the "it" refers to the equation (does the limit exist-type question).

Sorry again.

 

[mmm4444bot]

Well, I'm still confused about 10(a), but if 10(b) asks whether or not the limit of the posted rational function (as x approaches 2) exists, then look at what value the function is approaching as x approaches 2 from above AND as x approaches 2 from below.

If these two values are the same, then the limit exists.

The easiest way is to examine a proper graph of the rational function near x = 2. I'm not sure if 10(b) wants you to calculate the two directional limits by hand because you did not post the entire exercise.