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Chapter 11: Q. 5 (page 901)

Find the given limits if they exist. If a limit does not exist, explain why.
$\lim_{t 鈫� 0} (\frac{\frac{1}{(3 + t) - \frac{1}{3}}}{t}, \frac{{(3 + t)}^{2} - 9}{t})$ .

Short Answer

Expert verified

The limit exists. The solution is,

$(- \frac{1}{3}, 6)$ .

Step by step solution

01

Step 1. Given Information.

The limit is,

$\lim_{t 鈫� 0} (\frac{\frac{1}{(3 + t) - \frac{1}{3}}}{t}, \frac{{(3 + t)}^{2} - 9}{t})$ .

02

Step 2. Whether the limit exists or not.

Consider the limit,

$\lim_{t 鈫� 0} (\frac{\frac{1}{(3 + t) - \frac{1}{3}}}{t}, \frac{{(3 + t)}^{2} - 9}{t})$

The objective is to evaluate the above limit.

$\lim_{t 鈫� t_{0}} r (t) = \lim_{t 鈫� t_{0}} (x (t), y (t), z (t)) = (\lim_{t 鈫� t_{0}} x (t), \lim_{t 鈫� t_{0}} y (t), \lim_{t 鈫� t_{0}} z (t))$

Here, $\lim_{t 鈫� t_{0}} x (t), \lim_{t 鈫� t_{0}} y (t),$ and $\lim_{t 鈫� t_{0}} z (t)$ exist.

03

Step 3. Finding the limit.

$\lim_{t 鈫� 0} (\frac{1}{t} (\frac{1}{(3 + t)} - \frac{1}{3}), \frac{{(3 + t)}^{2} - 9}{t}) = \lim_{t 鈫� 0} (\frac{1}{t} (\frac{3 - 3 - t}{(3 + t)}), \frac{9 + t^{2} + 6 t - 9}{t}) = \lim_{t 鈫� 0} (\frac{1}{t} (\frac{- t}{3 + t}), \frac{t^{2} + 6 t}{t}) = \lim_{t 鈫� 0} ((\frac{- 1}{3 + t}), t + 6) = (\lim_{t 鈫� 0} (\frac{- 1}{3 + t}), \lim_{t 鈫� 0} (t + 6)) = (- \frac{1}{3}, 6)$

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Most popular questions from this chapter

Let $r_{1} (f)$ and $r_{2} (t)$ be differentiable vector functions with three components each. Prove that

\frac{d}{d t} (r_{1} (t) - r_{2} (t)) = r_{1}^{'} (t) - r_{2} (t) + r_{1} (t) - r_{2}^{'} (t) 鈭�

(This is Theorem 11.11 (c).)

For each of the vector-valued functions in Exercises 22鈥�28, find the unit tangent vector.

$r (t) = (\cos^{3} t, \sin^{3} t)$

What is the dot product of the vector functions $r_{1} (t) = (x_{1} (t), y_{1} (t)) and r_{2} (t) = (x_{2} (t), y_{2} (t)) ?$

In Exercises 19鈥�21 sketch the graph of a vector-valued function $r (t) = (x (t), y (t))$ with the specified properties. Be sure to indicate the direction of increasing values oft.

Domainlocalid="1649578696830" $t 鈮� 0, r (0) = (0, 3), r (1) = (- 2, 1), \lim_{t 鈫� 鈭�} x (t) = - 5, and \lim_{t 鈫� 鈭�} y (t) = - 鈭� .$

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.
$((\sin t) i + (\cos t) j + t k) 脳 ((\cos t) i + (\sin t) j)$

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