Limit of integer part function using Sandwich rule

• MHB
• Yankel
In summary, the conversation discusses calculating the limit of a function using the sandwich rule and the integer part function. The limit is found to be the constant a as x goes to infinity.
Yankel
Hello everyone,

I want to calculate the following limits:

$\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}$

using the sandwich rule, where [xa] is the integer part function defined here:

Integer Part -- from Wolfram MathWorld

I am not sure how to approach this. Any assistance will be most appreciated.

For the integer part of some $u$, we have:
$$u-1 < < u+1$$
That is, we are rounding either up or down, meaning that we end up less than 1 higher or lower.

So:
$$x\cdot a - 1 < [x\cdot a] < x\cdot a + 1$$
With positive $x$, it follows that:
$$\frac{x\cdot a-1}{x}<\frac{[x\cdot a]}{x}< \frac{x\cdot a + 1}{x} \implies a-\frac 1x < \frac{[x\cdot a]}{x} < a + \frac 1x$$
Now let $x$ go to infinity and apply the sandwich theorem.

Great idea...

so the correct answer is the constant a then ?

If x goes to infinity, 1/x goes to 0

Yankel said:
Great idea...

so the correct answer is the constant a then ?

If x goes to infinity, 1/x goes to 0

Yep. (Nod)

1. What is the integer part function?

The integer part function, denoted as [x], is a mathematical function that returns the largest integer less than or equal to a given number x. It essentially rounds down to the nearest integer.

2. What is the limit of the integer part function?

The limit of the integer part function does not exist, as the function is discontinuous at all non-integer values. However, we can still use the sandwich rule to find the limit at certain points.

3. What is the sandwich rule?

The sandwich rule, also known as the squeeze theorem, is a method for finding the limit of a function by comparing it to two other functions whose limits are known. If the two functions "squeeze" the original function, then the limit of the original function must also be the same as the limits of the two "squeezing" functions.

4. How do you use the sandwich rule to find the limit of the integer part function?

To use the sandwich rule for the integer part function, we can choose two functions that "squeeze" the function at a specific point. For example, if we want to find the limit of [x] as x approaches 2, we can choose the functions f(x) = x and g(x) = x+1. Both of these functions are greater than [x] at x=2, and their limits as x approaches 2 are also 2. Therefore, by the sandwich rule, the limit of [x] at x=2 is also 2.

5. What are some real-life applications of the integer part function and the sandwich rule?

The integer part function is often used in computer programming to round down numbers to the nearest integer. The sandwich rule is a useful tool in calculus for finding the limit of a function that may be difficult to evaluate directly. It can also be used in physics and engineering to approximate values and make predictions based on known limits.

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