- #1
Berrius
- 19
- 0
This is not really a homework question, but I've come across this while preparing for a test
Let [tex]f:U \subseteq R^n -> R^m[/tex] be a function which is differentiable at [tex]a \in U[/tex], and [tex]u \in R^n[/tex]
It is then stated that it is clear that:
[tex] lim_{t \to 0} \frac{||f(a+t*u)-f(a)-D_f(t*u)||}{||t*u||} = 0 => lim_{t \to 0} \frac{f(a+t*u)-f(a)-D_f(t*u)}{|t|} = 0[/tex]
How do they get this result?
I've tried using the epsilon/delta-def of limits but where the first limit is about real numbers, the second is about vectors in the R^m, so I am totally confused.
Homework Statement
Let [tex]f:U \subseteq R^n -> R^m[/tex] be a function which is differentiable at [tex]a \in U[/tex], and [tex]u \in R^n[/tex]
It is then stated that it is clear that:
[tex] lim_{t \to 0} \frac{||f(a+t*u)-f(a)-D_f(t*u)||}{||t*u||} = 0 => lim_{t \to 0} \frac{f(a+t*u)-f(a)-D_f(t*u)}{|t|} = 0[/tex]
How do they get this result?
The Attempt at a Solution
I've tried using the epsilon/delta-def of limits but where the first limit is about real numbers, the second is about vectors in the R^m, so I am totally confused.