Chapter 11: Q. 7 (page 901)

Finding limits: Find the given limits if they exist. If a limit does not exist, explain why.
$\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{1 / t}>$

Short Answer

Expert verified

Ans: $\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{\frac{1}{t}}> = ⟨ 1, 0, e ⟩$

Step by step solution

Step 1. Given information:

$\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{1 / t}>$

Step 2. FInding the limit of a vector function:

Definition of the limit of a vector function

Assume $r (t) = ⟨ x (t), y (t) ⟩$ be a vector valued function. The limit of $r (t)$ as $t$ approaches $t_{0}$ , denoted by $\lim_{t \to t_{0}} r (t)$ is defined by

$\lim_{t \to t_{0}} r (t) = \lim_{t \to t_{0}} ⟨ x (t), y (t), z (t) ⟩ = <\lim_{t \to t_{0}} x (t), \lim_{t \to t_{0}} y (t), \lim_{t \to t_{0}} z (t)>,$

where $\lim_{t \to t_{0}} x (t), \lim_{t \to t_{0}} y (t)$ and $\lim_{t \to t_{0}} z (t)$ exist .

Step 3.

Rewrite the limit,

$\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{\frac{1}{t}}> = <\lim_{t \to 0} \frac{\sin t}{t}, \lim_{t \to 0} \frac{1 - \cos t}{t}, \lim_{t \to 0} (1 + t)^{\frac{1}{t}}>$

$= <1, \lim_{t \to 0} \frac{1 - \cos t}{t}, e> S i n c e \lim_{t \to 0} \frac{\sin t}{t} = 1, \lim_{t \to 0} (1 + t)^{\frac{1}{'}} = e = <1, \lim_{t \to 0} \frac{\sin t}{1}, e> \{\begin{array}{l} \lim_{t \to 0} \frac{1 - \cos t}{t} i s \frac{0}{0} f o r m s o u s e L' H o s p i t a l r u l e, \\ \lim_{t \to 0} \frac{1 - \cos t}{t} = \lim_{t \to 0} \frac{\frac{d}{d t} (1 - \cos t)}{\frac{d}{d t} (t)} = \lim_{t \to 0} \frac{\sin t}{1} \end{array}\}$

$= ⟨ 1, 0, e ⟩$

Therefore, $\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{\frac{1}{t}}> = ⟨ 1, 0, e ⟩$

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Finding limits: Find the given limits if they exist. If a limit does not exist, explain why.
$\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{1 / t}>$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. FInding the limit of a vector function:

Step 3.

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Finding limits: Find the given limits if they exist. If a limit does not exist, explain why.limt→0sintt,1-costt,(1+t)1/t

Short Answer

Step by step solution

Step 1. Given information:

Step 2. FInding the limit of a vector function:

Step 3.

One App. One Place for Learning.

Most popular questions from this chapter

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Finding limits: Find the given limits if they exist. If a limit does not exist, explain why.
$\lim_{t \to 0} <\frac{\sin t}{t}, \frac{1 - \cos t}{t}, (1 + t)^{1 / t}>$