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Stewart, page 800 says:

"Proof: If we define a function g of the single variable h by

[tex] g(h) = f(x_0 + ha, y_0 + hb) [/tex]

then by the definition of a derivative we have

[tex] g'(0)= lim_{h \rightarrow 0} \frac{g(h) - g(0)}{h} = lim_{h \rightarrow 0} \frac{f(x_0+ha, y_0+hb)-f(x_0,y_0)}{h} [/tex]

end quote

Is it me, or is he over using the variable h? He defines a function called g(h). And then he puts h back into the derivative. if h=0, then it does not make sense to say g(h)-g(0), becuase he said before that h=0. Should he call the function g(h), g(h'), and then he can call the h in the limit, plain old h?