Limit at \(0^+\)

Calculus Level pending

Consider the function \(f(x)=\frac{1}{x}\). For each positive value of \(h\), define \( \theta_h\) to be the unique positive value that satisfies the equation \[ f( 2 + h) = f(2) + h f'(2 + \theta_h h ). \] What is the value of \(\displaystyle \lim_{h \to 0^+} \frac{1}{\theta_h}\)?

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