Proving Inverse Function Continuity at Limit Point Q

[STEMucator]

Homework Statement

Suppose f is a function defined on a set $S$ in $ℝ^n$ and suppose $Q$ is a limit point of $S$.

If $f(P) → 3$ as $P → Q$ prove from first principles that $\frac{1}{f(P)} → \frac{1}{3}$ as $P → Q$.

Homework Equations

The Attempt at a Solution

I'm a bit rusty with these.

I know : $\forall ε'>0, \exists δ'>0 \space | \space 0 < |P-Q| < δ' \Rightarrow |f(P) - 3| < ε'$

I want : $\forall ε>0, \exists δ>0 \space | \space 0 < |P-Q| < δ \Rightarrow |1/f(P) - 1/3| < ε$

For some reason I'm blanking on what to do next.

 

[tiny-tim]

HI Zondrina!

Hint: |1/f(P) - 1/3| = |f(P) - 3| / |f(P)|

 

[STEMucator]

HI Zondrina!

Hint: |1/f(P) - 1/3| = |f(P) - 3| / |f(P)|

Thanks tim, cleaned up nicely :)