- #1
H2Pendragon
- 17
- 0
Homework Statement
Use the definition of limits to show that
[tex]lim _{x\rightarrow 2} \frac{1}{x^{2}} = \frac{1}{4}[/tex]
Homework Equations
[tex]\forall \epsilon > 0, \exists \delta > 0, x \in D, 0 < |x-2| < \delta \Rightarrow |\frac{1}{x^{2}} - \frac{1}{4}| < \epsilon [/tex]
The Attempt at a Solution
[tex]|\frac{1}{x^{2}} - \frac{1}{4}| < \epsilon \Rightarrow |\frac{4-x^{2}}{4x^{2}}| < \epsilon \Rightarrow \frac{|4-x^{2}|}{4x^{2}} < \epsilon [/tex]
[tex]Restrict: 1 < x < 3 \Rightarrow 4<4x^{2}<36 \Rightarrow \frac{1}{36} < \frac{1}{4x^{2}} < \frac{1}{4} \Rightarrow \frac{|4-x^{2}|}{36} < \frac{|4-x^{2}|}{4x^{2}} < \frac{|4-x^{2}|}{4} < \epsilon[/tex]
And now I'm stuck. I'm trying to find delta based on epsilon. Normally the problems come to a neat little solution, but |4-x2| is not |x-2| and trying to get it to equal |x-2| is where I'm having the trouble.
Any help is appreciated.