# Prove a limit using inequalities

• Hernaner28
In summary, using the given inequalities, we can prove that the limit of x/sinx as x approaches 0 is equal to 1, by showing that for every epsilon > 0, there exists a delta > 0 such that for all x, |x-0| < delta implies |x/sinx - 1| < epsilon. This can be achieved by assigning a specific value to delta in terms of epsilon and using the given inequalities to manipulate the expression.
Hernaner28
Using the inequalities: $$\sin x \leq x \leq \tan x$$ valid in a zero range, prove that:

$$\displaystyle\lim_{x \to{0}}{\frac{x}{\sin x}}= 1$$

Thank you!

I'm rather unsure about this, but I'm deeply interested in the answer. I have attempted a solution that may or may not be correct.

What we're trying to prove is that for every epsilon > 0, there is some delta > 0 such that for all x, |x- 0| < delta and |x/sinx - 1| < epsilon. Because of this, we can assign a delta a particular value in terms of the epsilon.

if sinx ≤ x ≤ tanx, then (1/sinx) ≥ (1/x) ≥ (1/tanx)

Factor |(x/sinx) - 1| into |x| * |(1/sinx - 1/x)|

|(1/sinx - 1/x)| ≤ |1/sinx| + |1/x| < |1/sinx| + |1/sinx| by a manipulation of our given inequality = |2/sinx|

Let δ = min(1, ε/|2/sinx|)
from |x| < δ
we have |x| < ε/|2/sinx|
|x|*|(1/sinx + 1/x)| < ε/|2/sinx|*|(1/sinx + 1/x)|
|x/sinx - 1| < ε/|2/sinx|* |2/sinx| by inequality above
|x/sinx - 1| < ε

Last edited:

## 1. How do you prove a limit using inequalities?

To prove a limit using inequalities, you need to show that for any given value of epsilon (ε), there exists a corresponding value of delta (δ) such that if the distance between the input and the limit is less than delta, then the distance between the output and the limit is less than epsilon.

## 2. What is the importance of using inequalities in proving a limit?

Inequalities are important in proving a limit because they allow us to establish a clear relationship between the input and output values, and determine the behavior of a function near a certain point. They also help us to show that a limit exists and is unique.

## 3. Can you give an example of proving a limit using inequalities?

For example, to prove that the limit of the function f(x) = 2x + 1 as x approaches 2 is 5, we can use the following inequality: |2x + 1 - 5| < ε, where ε is a small positive number. By rearranging the inequality, we get |x - 2| < ε/2. This shows that for any ε, we can choose δ = ε/2 to satisfy the definition of a limit.

## 4. What are some common techniques used in proving limits using inequalities?

Some common techniques used in proving limits using inequalities include using the definition of a limit, manipulating and simplifying inequalities, and using the triangle inequality to combine multiple inequalities.

## 5. Is using inequalities the only way to prove a limit?

No, using inequalities is not the only way to prove a limit. Other methods such as the squeeze theorem, direct substitution, and the epsilon-delta definition can also be used to prove limits. However, using inequalities is a widely used and effective method for proving limits in many cases.

Replies
3
Views
561
Replies
5
Views
742
Replies
10
Views
1K
Replies
2
Views
501
Replies
2
Views
1K
Replies
2
Views
1K
Replies
19
Views
711
Replies
12
Views
2K
Replies
13
Views
408
Replies
13
Views
1K