Prove by using the formal definition of limit that limx--->∞ 1/(x^4+x^2+5)=0

Question

This is what I have so far:

 

Limx-->∞ 1/(x^4+x^2+5)=0 if for all ϵ>0 we find a value x_c depending on ϵ such that | 1/(x^4+x^2+5)|< ϵ

 

I dont know what to do afterwards

Answer 1

That answer is a bit confused!

The correct definition is: limit as x → ∞ f(x) = L iff, given ε > 0, ∋ x_c = x_c(ε) s.t. x > x_c ⇒ |f(x) - L| < ε

In other words, by taking x sufficiently large we can make f(x) as close to L as we wish.

So, for the given function, f(x) = 1/(x⁴ + x² +5), we need to find that special x_c. And here, the limit L = 0.

Given arbitrary ∈ > 0, suppose |f(x) - 0| < ∈ ⇒ |1/(x⁴ + x² +5)| < ε

⇔ 1/(x⁴ + x² +5) < ε since x⁴ + x² + 5 > 0 for all x ∈ R

⇔ x⁴ + x² +5 > 1/ε

⇔ x⁴ + x² > 1/ε - 5

Choose x_c > max(1/∈ - 5, 1). Then, x⁴ > x >x_c and x² > x > x_c ⇒ x⁴ + x² +5 > 1/ε

⇒ |x⁴ + x² +5| > 1/ε

⇒ |1/(x⁴ + x² +5)| < ε