find the limit: (lim_{t\rightarrow0}leftlangle\frac{e^{7t}-1}{t},\frac{t^{6}}{t^{7}-t^{6}},\frac{2}{16 +…

Explanation:

Step1: Find limit of first - component

Recall the limit formula $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}=1$. Let $x = 7t$. As $t\rightarrow0$, $x\rightarrow0$. So, $\lim_{t\rightarrow0}\frac{e^{7t}-1}{t}=7\lim_{t\rightarrow0}\frac{e^{7t}-1}{7t}=7\times1 = 7$.

Step2: Find limit of second - component

Factor the denominator: $\lim_{t\rightarrow0}\frac{t^{6}}{t^{7}-t^{6}}=\lim_{t\rightarrow0}\frac{t^{6}}{t^{6}(t - 1)}$. Cancel out $t^{6}$ (since $t\neq0$ when taking the limit), we get $\lim_{t\rightarrow0}\frac{1}{t - 1}=\frac{1}{0 - 1}=-1$.

Step3: Find limit of third - component

Substitute $t = 0$ into $\frac{2}{16 + t}$. We have $\lim_{t\rightarrow0}\frac{2}{16 + t}=\frac{2}{16+0}=\frac{1}{8}$.

Answer:

$\left\langle7,-1,\frac{1}{8}\right\rangle$