Calculus Troubles: Finding Limits and Understanding 0/0

[cvc121]

Homework Statement

Find the following limits, if they exist.

Homework Equations

The Attempt at a Solution

I have just started calculus and am having trouble with 3 a). I get 0/0 after substitution so I factored but still get 0 in the denominator. Does this indicate that the limit does not exist? Am I doing the questions correctly? My work is attached below. Thanks! All help is very much appreciated.
View attachment 96417https://www.physicsforums.com/attachments/96417/

 

[Gianmarco]

You're taking the limit of a constant (3) divided by something which approaches zero. What do you get if you divide a positive constant by an infinitesimal quantity?

 

[cvc121]

Undefined = DNE?

 

[cvc121]

Or would it be positive infinity?

 

[vela]

You could also consider the graph of $\frac{x+2}{x-1}$. It should be clear from that that the limit doesn't exist.

 

[Gianmarco]

Or would it be positive infinity?

Yes, but you can approach 1 from both sides. $lim_{x\rightarrow 1^+} \frac{x+2}{x-1}$ must be equal to $lim_{x\rightarrow 1^-} \frac{x+2}{x-1}$ for the limit to exist. Are they?

 

[HallsofIvy]

Saying that a limit "is positive infinity" is just saying that "the limit does not exist" but giving a specific reason- that the values get larger and larger and larger without upper bound, as opposed to getting lower and lower without lower bound or jumping around without getting close to anyone specific number.

 

[Ray Vickson]

Saying that a limit "is positive infinity" is just saying that "the limit does not exist" but giving a specific reason- that the values get larger and larger and larger without upper bound, as opposed to getting lower and lower without lower bound or jumping around without getting close to anyone specific number.

Very nice explanation.