I was reviewing my MV Calculus (got a bit rusty) and was stuck at this:
Question 1
If I have a function (curve) in $N$ dimensional space which is parametricized as $g(t)$. Now, g'(t)|_{t=a} gives the tangent vector at $t=a$.
The book states (without proof) that the tangent line to the curve is given by the set \{g(a) + t\times g'(a)\ | \text{ }t \in \mathbb{R} \}
How is this true? Isn't the tangent vector simply g'(t)|_{t=a}
Consider g(t) = $\begin{pmatrix} sin(t)\\ cos(t)\\ t \end{pmatrix}$, g'(t) = $\begin{pmatrix} cos(t)\\ -sin(t)\\ 1 \end{pmatrix}$
At $t=\frac{\pi}{2}$, g'(t) = $\begin{pmatrix} 0\\ -1\\ 1 \end{pmatrix}$
What ahead?
Question 2
Is $\lim_{x \to a} a^{f(x)}$ the same as $a^{\lim_{x \to a} {f(x)}}$ ?
a is a constant