Limits Confusion: What Is Wrong?

[zorro]

What is wrong?

 

[CompuChip]

Nothing's wrong, the fact that you can rewrite the function such that the limit becomes of the form 'infinity/infinity' when evaluated 'naively' does not mean that the limit suddenly no longer exists. In fact, this is one of the cases where we are allowed to apply l'hospital's rule, and we then often find a finite answer.

What you could also do, of course, is rewrite it as
$$\lim_{n \to \infty} \frac{1 + x^n}{1 + x^n} - \frac{2 x^n}{1 + x^n}$$
and then divide numerator and denominator of the second term by xⁿ (or you can still do that first and then do the same rewriting trick, it just looks a little different).

 

[dextercioby]

I hope the OP sees that nothing in his limit diverges so would it need to be <reingeneered> ??

 

[zorro]

I hope the OP sees that nothing in his limit diverges so would it need to be <reingeneered> ??

Re-engineering helps to understand things better.

Nothing's wrong, the fact that you can rewrite the function such that the limit becomes of the form 'infinity/infinity' when evaluated 'naively' does not mean that the limit suddenly no longer exists. In fact, this is one of the cases where we are allowed to apply l'hospital's rule, and we then often find a finite answer.

What you could also do, of course, is rewrite it as
$$\lim_{n \to \infty} \frac{1 + x^n}{1 + x^n} - \frac{2 x^n}{1 + x^n}$$
and then divide numerator and denominator of the second term by xⁿ (or you can still do that first and then do the same rewriting trick, it just looks a little different).

Thanks!

 

[Landau]

$$1=\lim_{n\to\infty}(1)=\lim_{n\to\infty}\frac{n}{n}\text{'}=\text{'}\frac{\infty}{\infty}$$

You are applying the quotient rule where it is invalid to do so. The quotient rule says that IF $$\lim_{n\to\infty}a_n=a$$ and $$\lim_{n\to\infty}b_n=b$$, where a and b are real (FINITE!) numbers with b non-zero, THEN $$\lim_{n\to\infty}\frac{a_n}{b_n}=\frac{a}{b}$$.

In this case a and b are not (finite) real numbers.

 

[zorro]

Is CompuChip wrong?