Question

Math

Posted 4 months ago

```
We want to find $\lim _{x \rightarrow 0} \frac{x}{\sin (x)}$. Direct substitution and other algebraic methods don't seem to work.
Looking at the graph of $f(x)=\frac{x}{\sin (x)}$, we can estimate that the limit is equal to 1 .
To prove that $\lim _{x \rightarrow 0} \frac{x}{\sin (x)}=1$, we can use the squeeze theorem.
Pablo suggested that we use the functions $g(x)=-x^{2}+0.5$ and $h(x)=x^{2}+1.5$ in order to apply the squeeze theorem.
Does Pablo's suggestion seem to be correct?
Choose 1 answer:
(A) Yes, Pablo's suggestion seems to be correct.
(B) No, Pablo's suggestion is incorrect because it's not true that one function is always below $f$ and one function is always above it for $x$-values near 0 .
(c) No, Pablo's suggestion is incorrect because it's not true that the limits of $g$ and $h$ are both equal to 1 .
```

Sign in to unlock the answer

Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

Looking at the graph of $f(x)=\frac{x}{\sin (x)}$, we can estimate that the limit is equal to 1

step 3

To prove that $\lim _{x \rightarrow 0} \frac{x}{\sin (x)}=1$, we can use the squeeze theorem

step 4

Pablo suggested that we use the functions $g(x)=-x^{2}+0.5$ and $h(x)=x^{2}+1.5$ in order to apply the squeeze theorem

step 5

The limits of $g(x)$ and $h(x)$ as $x$ approaches 0 are $\lim_{x \to 0} g(x) = 0.5$ and $\lim_{x \to 0} h(x) = 1.5$

step 6

Since the limits of $g(x)$ and $h(x)$ are not both equal to 1, Pablo's suggestion is incorrect

C

Key Concept

Squeeze Theorem

Explanation

The squeeze theorem requires that the limits of the bounding functions be equal to the limit of the function in question. Since the limits of $g(x)$ and $h(x)$ are not both equal to 1, Pablo's suggestion is incorrect.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question