Find log_x(4) as x-->1: Understanding the Limit of logx(4)

[strigner]

Homework Statement
Evaluate
lim x-->1 log_x(4)


The attempt at a solution
I can't understand this because basically if you plug in 1 as x, log₁4 doesn't have a solution because 1 to the power of anything is just 1.

 

[HallsofIvy]

So that is clearly not a continuous function. But you don't find a limit by "plugging in" a number. If y= log_x(4), then 4= x^y. I think what I would do is let h= x-1 so that 4= (1+h)^y and the limit is as h goes to 0. Apply the generalized binomial formula to (1+ h)^y

 

[strigner]

Thanks for the response, now I remember that I'm not looking for the value at 1, so that makes sense now. But I'm not sure what to do with that binomial as we haven't covered binomial formula.

I tried thinking about it with RHS and LHS limits. If you're approaching 1 from the left then y is a large negative and if you are approaching from the right then y is a large positive number, therefore the limit at 1 does not exist. Does that make sense?

 

[statdad]

HallsofIvy, correct me if I'm off track, but I think your approach hides the dependence of the limit in this variable $$y$$, and since there is no simple way to evaluate the generalized series, I'm not sure where to turn.

My first thought was this:

$$\log_x {4} = \frac{\log 4}{\log x}$$

(I used logs base 10; obviously $$\ln$$ would also do)
Then look at the limit of this expression from the right and left. Explorations with SAGE confirmed my suspicions about this.

If I've missed something obvious, please let me know.

 

[strigner]

So even with $$\log_x {4} = \frac{\log 4}{\log x}$$ it looks like the limit does not exist because when approaching 1 from the left the denominator is a very small negative number making the limit negative infinity and when approaching from the right the denominator is a very small positive number making the limit postiive infinity, therefore limit does not exist, is that correct?

 

[Gib Z]

Yes.

 

[HallsofIvy]

Actually that was my point. Using the generalized Binomial formula for (1+ h)^y gives 1+ hy+ higher power terms in h. 4= 1+ hy+ higher power terms in h. Taking the limit as h goes to 0 gives 4= 1 no matter what y is: the limit does not exist.