Solved: "Find the limit of the following vector-valued functions..."

Ashley Searcy

Answered question

2021-11-28

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}) ⟩

Answer & Explanation

Lorraine David

Beginner2021-11-29Added 13 answers

Step 1
Given that the vector valued function and we need to find the limit of the following vector - valued functions.
The vectored valued function is given by
$r (t) = lim_{t \to 4} [\sqrt{t - 3}, \frac{\sqrt{t} - 2}{t - 4}, \tan (\frac{π}{t})]$
Step 2
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 ofthe 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
$= [lim_{t \to 4} \sqrt{t - 3}, lim_{t \to 4} \frac{1}{2 \sqrt{t}}, lim_{t \to 4} \tan \frac{π}{t}]$
$[\sqrt{4 - 3}, \frac{1}{2 \sqrt{4}}, \tan \frac{π}{4}]$
$= [1, \frac{1}{2 ⋆ 2}, 1]$
$= [1, \frac{1}{4}, 1]$

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