Vector Functions — Question 6

Question 6 – Solution

Strategy A vector limit equals the target precisely when all component limits equal their corresponding target components.

First component Rationalizing gives $$\frac{\sqrt{1+at}-1}{t}=\frac{a}{\sqrt{1+at}+1}\longrightarrow\frac a2,$$ so $a=4$.

Other components Standard limits give $\sin(bt)/t\to b$ and $(e^{ct}-1)/t\to c$. Hence $\boxed{a=4,\ b=3,\ c=-4}$.

Verification The three component limits are respectively $2,3,-4$.