Exercise · ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-010 · ★★★ Advanced
For the same \(f\) (with \(f_x(0,0)=f_y(0,0)=0\)), what value does the error quotient \(\dfrac{f(h,k)}{\sqrt{h^2+k^2}}\) approach along the line \(h=k>0\)?
Choose one option.
A \(0\) B \(\dfrac{\sqrt2}{4}\) C \(\dfrac12\) D It does not exist Show answer ← Back to the question
Explanation With \(h=k\): \(f(h,h)=h/2\) and \(\sqrt{h^2+h^2}=\sqrt2\,h\), so the quotient is \(\frac{1}{2\sqrt2}=\frac{\sqrt2}{4}\), constant and nonzero.
A function \(f(x,y)\) is differentiable at a point \((a,b)\) if its graph has a tangent plane there: the error of the linear approximation is negligible compared with the distance to the point.
\[\lim_{(h,k)\to(0,0)}\frac{f(a+h,b+k)-f(a,b)-f_x(a,b)\,h-f_y(a,b)\,k}{\sqrt{h^2+k^2}}=0\]
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