I am reading a text where directional derivatives of functions $f:E\rightarrow F$, where $E,F$ are Banach spaces, at the point $x_0 \in E$ are defined as $d_v f(x_0)=\lim_{t\rightarrow 0+} \frac{f(x_0+tv)-f(x_0)}{t}$ for any $v\in E$. My question is: Why is the limit taken with respect to "$t\rightarrow 0+$" instead of just "$t\rightarrow 0$" ? What does it change if we have "$t\rightarrow 0+$ instead of "$t\rightarrow 0$" ? Since in some textbooks, just the ""$t\rightarrow 0$" version is used.
Ways to define the directional derivative
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analysis
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0sorry for the very late response...@3Sphere http://mathworld.wolfram.com/DirectionalDerivative.html ; there $\lim_{ t \rightarrow 0}$ instead of $\lim_{ t \rightarrow 0+}$ is permitted @gary no, since $t$ is just a scalar, which I can use to divide. I need norms only when defining the differential, I think, not the directional derivative @Dactyl / Christian Blatter Could you maybe provide an examples where the directional derivative at a point on the boundaries of the domain for $t \rightarrow 0$ exists, but for$ t \rightarrow 0+$ doesn't / where $\lim_{ t \rightarrow 0}$ isn't linear ? – 2011-06-22