# What am I missing here in this difference quotient problem?

Gold Member
I've highlighted the part in yellow I don't understand. He apparently 'drops' Δx in the last line, but doesn't display how. What I do know is that he is taking the limit as Δx→0, or as Delta x approaches zero. I'm simpy missing what he did though to drop the Δx. I mean, I understand he distributed x into the parentheses to get x0 squared, but not how he just dropped the Δx.

Any insight?

symbolipoint
Homework Helper
Gold Member
The work on the chalkboard looks good. The first pair of delta x which the lecturer canceled acted as a factor of 1. The highlighted delta x would become increasingly, vanishingly small and so is taken as near enough to zero. The lecturer has shown the limit of the original expression as delta x approaches zero.

(minor edit)

Last edited:
phinds
Gold Member
2021 Award
He did not drop anything. He DID do a cross-cancellation of a delta X on top and a delta X on bottom. He even shows lines through them to emphasize this.

If you are asking about the very last bit of "= -1 / (X0) ^ 2 then that is, as he clearly states, a limit as delta X approaches zero.

Gold Member
The highlighted delta x sub zero would become increasingly, vanishingly small and so is taken as near enough to zero. The lecturer has shown the limit of the original expression as delta x sub zero approaches zero.

Does that mean that since delta x is approaching zero, it is just left out because it is so small? I guess I just don't understand at all.

phinds
Gold Member
2021 Award
Does that mean that since delta x is approaching zero, it is just left out because it is so small? I guess I just don't understand at all.
Yes, that is exactly what it means. The concept of the limit is that you see what would happen if it actually GOT to the limiting amount.

symbolipoint
Homework Helper
Gold Member
Does that mean that since delta x is approaching zero, it is just left out because it is so small? I guess I just don't understand at all.
Yes, that is exactly what it means. The concept of the limit is that you see what would happen if it actually GOT to the limiting amount.
Understand that in the shown example, letting delta x become zero does not make the denominator part of the expression undefined.

(Reminder: I made a minor edit to my response.)

micromass
Staff Emeritus