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vousetmoiec

Answered question

2021-11-26

Find the limit of the following vector-valued functions at the indicated value of t.

lim_{t \to 4} ⟨ \sqrt{t - 3}, \frac{\sqrt{t -} 2}{t - 4}, \tan (\frac{π}{t}) ⟩

Vincent Diaz

Beginner2021-11-27Added 19 answers

Step 1
To find the limit , we evaluate the limit of each component

lim_{t \to 4} r (t) = lim_{t \to 4} [\sqrt{t - 3}, \frac{\sqrt{t} - 2}{t - 4}, \tan \frac{π}{t}]

= [lim_{t \to 4} \sqrt{t - 3}, lim_{t \to 4} \frac{\sqrt{t} - 2}{t - 4}, lim_{t \to 4} \tan \frac{π}{t}]

since second term when applied limit is of the form

\frac{0}{0}

we use L hospital rule. So

\frac{\sqrt{t} - 2}{t - 4}

will be equal to

\frac{\sqrt{t} - 2}{t - 4} = \frac{\frac{1}{2 \sqrt{t}}}{1}

- \frac{1}{2 \sqrt{t}}

So r(t) changes to

= [lim_{t \to 4} \sqrt{t - 3} lim_{t \to 4} \frac{1}{2 \sqrt{t}}, lim_{t \to 4} \tan \frac{π}{t}]

= [\sqrt{4 - 3}, \frac{1}{2 \sqrt{4}}, \tan \frac{π}{4}]

= [1, \frac{1}{2 \times 2}, 1]

= [1, \frac{1}{4}, 1]

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