- #1
ninty45
- 4
- 0
Homework Statement
Given the piecewise function
[itex]
f(x) = \left\{
\begin{array}{lr}
\frac{(2-x)^2-p}{x} &: x < q\\
r(x+6) &: q \leq x <2 \\
x^3-p &: x \geq 2
\end{array}
\right.
[/itex]
Find the values of p,q,r such that f(x) is continuous everywhere and f(2) = p
The Attempt at a Solution
Since f(2) = p,
[itex]
f(2) = (2)^3 - p = p\\
8-p=p\\
p=4
[/itex]
If f(x) is continuous everywhere, then
[itex]
\lim_{x \to 2} f(x) = f(2)
[/itex]
But f(2)=p, and r(x+6) is always smooth, hence
[itex]
r(2+6) = 4 \\
r = \frac{1}{4}
[/itex]
My difficulty is with q. Again from (assumed) continuity, at x=q,
[itex]
\lim_{x \to q} f(x) = f(q) \\
\lim_{x \to q} \frac{(2-x)^2-4}{x} = \frac{1}{2}(q+6) \\
\lim_{x \to q} x-4 = \frac{1}{2}(q+6) \\
2q-8 = q+6 \\
q=14
[/itex]
but from the original function, q <2. The answer given is supposedly q=0, but I do not see then how the function could be continuous.
Are my calculations wrong? I tried assuming that q =0, hence instead of factorising the limit I used l'hopital but no dice.
I am including the original picture as a comparison.