First Principles-( ε-δ methods) proof.

[manooba]

(X^2+1)/(X^3-9) → -1/4 as x → 1

(without using algebra of limits. Also constructing δ explicitly).

I have attempted this question but to no avail,

I start like this:

= |(X^2+1)/(X^3-9)+1/4|

= (4(X^2+1)+X^3-9)/ 4(X^3-36)

try factorisation but am unsuccessful.

Anybody lend a helping hand?

 

[SammyS]

(X^2+1)/(X^3-9) → -1/4 as x → 1

(without using algebra of limits. Also constructing δ explicitly).

I have attempted this question but to no avail,

I start like this:

= |(X^2+1)/(X^3-9)+1/4|

= (4(X^2+1)+X^3-9)/ 4(X^3-36)

try factorisation but am unsuccessful.

Anybody lend a helping hand?

Expand (multiply out) the terms in the numerator. Then collect like terms and factor the result.

That's a start.