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Chapter 1: Problem 57

Finding a Limit What is the limit of \(f(x)=4\) as \(x\) apprraches \(\pi ?\)

Short Answer

Expert verified
The limit of \(f(x)=4\) as \(x\) approaches \(\pi\) is 4.

Step by step solution

01

Identify the constant function

Here, it is given that \(f(x)=4\). This is a constant function, where the output (or the y-value) is 4 no matter what the input or the x-value is.
02

Evaluate the Limit

To find the limit as \(x\) approaches any value, look at the value of the function. Because the function is constant, the value of the function as \(x\) approaches \(\pi\) is 4. Thus, the limit of \(f(x)=4\) as \(x\) approaches \(\pi\) is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Function
A constant function is one of the simplest forms of mathematical functions. It's called "constant" because it always gives the same output, regardless of the input. For instance, in our exercise, the function is given by \( f(x) = 4 \). This means that for any value of \( x \), whether it's 0, 1, -5, or even \( \pi \), the function will always return 4.
  • In a constant function, the "y" value remains unchanged.
  • This function can be visualized as a horizontal line on a graph, intersecting the y-axis at the constant value, in this case, 4.
  • Constant functions are fundamental in understanding limits because they show predictable patterns.
Overall, constant functions are elementary but powerful tools in calculus, providing a straightforward way to deal with many limit-related problems.
Evaluate Limit
Evaluating a limit involves determining what value a function approaches as the input gets closer to a certain point. For the constant function \( f(x) = 4 \), this is particularly simple. Since the value of \( f(x) \) is always 4, its limit, as \( x \) approaches any value, be it \( \pi \) or something else, remains 4.
Here is how you can evaluate it:
  • Identify the limit point (where \( x \) is approaching). In our example, \( x \) approaches \( \pi \).
  • Look at the function's output. Since it is a constant function, the result will always be 4, no matter the approaching value.
  • Conclude that the limit of the function \( f(x) \) as \( x \to \pi \) is simply 4.
This concept simplifies the practice of finding limits, especially when dealing with constant functions.
Approaching a Value
The concept of "approaching a value" in the context of limits is about understanding how a function behaves as its input (or \( x \)-coordinate) nears a specific number. In our exercise, we're interested in what happens to \( f(x) = 4 \) as \( x \) approaches \( \pi \).
For simpler functions like this constant function:
  • The function's output does not change as \( x \) approaches any point.
  • Essentially, the question becomes: "What value is the function stuck at?" Here, it's always 4.
  • Even though \( x \) is nearing \( \pi \), the proximity to \( \pi \) doesn't affect the outcome of \( f(x) \), which is consistently 4.
When understanding limits, recognizing that the behavior of constant functions remains unchanged, is crucial. The destination (approaching \( \pi \)) might change, but the value of the journey (the function's output) remains the same, offering a predictable and consistent limit of 4.

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