Suppose that f is an invertible function, f (0) = 0, f iscontinuous at 0, and
Chapter 1, Problem 56(choose chapter or problem)
Suppose that \(f\) is an invertible function, \(f (0) = 0\), \(f\) is continuous at \(0\), and \(\lim _{x-0}(f(x) / x)\) exists. Given that \(L=\lim _{x-0}(f(x) / x)\), show
\(\lim \limits_{x \rightarrow 0} \frac{x}{f^{-1}(x)}=L\)
[Hint: Apply Theorem 1.5.5 to the composition \(h◦g\), where
\(h(x)=\left\{\begin{array}{ll} f(x) / x, & x \neq 0 \\ L, & x=0 \end{array}\right.\)
And \(g(x)=f^{-1}(x)\). ]
Equation Transcription:
f
f (0) = 0
f
0
(f(x)/x)
L = (f(x)/x)
= L
hg
{
g(x) =
Text Transcription:
f
f (0) = 0
f
0
lim_x right arrow 0 (f(x)/x)
L= lim_x right arrow 0 (f(x)/x)
lim_x right arrow 0 x/f^-1 (x) = L
h circ g
h (x) = {_L, x = 0 ^f (x) / x, x neq 0
g(x) = f^-1 (x)
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