What is the Pedagogical Justification for Substitution?

Question

My mom is re-learning calculus for the third time (She tutors High School students). And she asked me a question invovling limits: $$\lim_{x\rightarrow \infty} \frac{\ln 2x}{\ln 3x}$$ I led her through my solution which was as follows: $$\lim_{x\rightarrow \infty} \frac{\ln 2x}{\ln 3x} = \lim_{x\rightarrow \infty} \frac{\ln 2 + \ln x}{\ln 3 + \ln x} =1 $$ When she didn't like that (the final jump), I tried: $$ \lim_{x\rightarrow \infty} \frac{\ln 2 + \ln x}{\ln 3 + \ln x} = \lim_{t\rightarrow \infty} \frac{\ln 2 + t}{\ln 3 + t} = 1$$

This seemed to be more respectable to her, but still she questioned the substitution $$t \gets \ln x$$

I used here. I didn't have a ready answer for how and why and when-it-is-OK for substitutions in general. I thought I'd ask here if someone can point me to a better (pedagogically better) way to explain the use of substitutions in limits (or derivatives or integrals--which are both limits). Thanks.

Answer 1

To state it explicitly: The substituted limit expression should have been $$ \lim_{e^t\to\infty} \frac{\ln(2)+ t}{\ln(3)+t} $$ but was replaced with $$ \lim_{t\to\infty} \frac{\ln(2)+ t}{\ln(3)+t} $$ I have not made up my mind yet if this will work with arbitrary expressions.

Answer 2

write it in the form $$\frac{\frac{\ln(2)}{\ln(x)}+1}{\frac{\ln(3)}{\ln(x)}+1}$$

Answer 3

If we take for granted that $$\lim_{t\rightarrow \infty} \frac{\ln 2 + t}{\ln 3 + t} = 1$$

then what this means is by making $t$ large enough, say greater than $x_0$, the fraction can be can be made within $\epsilon$ of $1$. Now, consider

$$\frac{\ln 2 + \ln x}{\ln 3 + \ln x}$$

When $\ln x$ is greater than $x_0$, this fraction will be within $\epsilon$ of $1$. If $\ln c = x_0$ , all $x>c$ will satisfy $\ln x > x_0$ since $\ln$ is increasing. We can thus claim that $$\lim_{x\rightarrow \infty} \frac{\ln 2 + \ln x}{\ln 3 + \ln x} = 1$$

If your mother does not know $\epsilon-\delta$, I don't think it is that hard to rephrase what I've written above in more intuitive language.

For "well behaved" functions which are continuous, monotone increasing and take on arbitrarily large values, these substitutions are entirely valid.