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Chapter 5: Problem 1

Prove the existence of the limit \(\lim _{x \rightarrow x_{0}}(4-12 x)\).

Short Answer

Expert verified
The limit exists and equals \(4 - 12x_0\).

Step by step solution

01

Identify the Definition of Limit

The limit of a function as it approaches a specific value is given by the expression \[ \lim_{x \rightarrow x_0} f(x) = L \]if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - x_0| < \delta \), we have \( |f(x) - L| < \epsilon \). This definition will guide us in proving the limit.
02

Substitute the Function and Point

We want to find and prove \( \lim_{x \rightarrow x_0} (4 - 12x) \). Substitute the function \( f(x) = 4 - 12x \) and evaluate the limit at point \( x_0 \).
03

Evaluate the Function Directly at \(x_0\)

To find \( \lim_{x \rightarrow x_0} (4 - 12x) \), substitute \( x_0 \) into the function: \[ f(x_0) = 4 - 12x_0 \].
04

Apply the Limit Definition

According to the limit definition, for the function's limit to exist and equal \( L = 4 - 12x_0 \), for every \( \epsilon > 0 \), we need to find a \( \delta > 0 \) such that \[ |(4 - 12x) - (4 - 12x_0)| < \epsilon \]when \( 0 < |x - x_0| < \delta \).
05

Simplify the Expression

We have, \[ |(4 - 12x) - (4 - 12x_0)| = |-12x + 12x_0| = 12|x - x_0| \]. We want \[ 12|x - x_0| < \epsilon \]to hold. This implies \[ |x - x_0| < \frac{\epsilon}{12} \].
06

Choose \(\delta\)

For the given \( \epsilon > 0 \), we choose \( \delta = \frac{\epsilon}{12} \). Thus, whenever \( 0 < |x - x_0| < \delta \), it follows that \[ 12|x - x_0| < \epsilon \].
07

Conclusion

Since for any \( \epsilon > 0 \), we found a \( \delta = \frac{\epsilon}{12} \) that meets the condition, the limit \[ \lim_{x \rightarrow x_0} (4 - 12x) = 4 - 12x_0 \]exists.

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Key Concepts

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

Epsilon-Delta definition
The Epsilon-Delta definition is a formal way to state what it means for a function to have a limit at a particular point. It ensures that as we get closer to a specific point, the values of the function get close to a certain number, called the limit. Here鈥檚 a simple way to understand it:
  • Epsilon (\( \epsilon \)): This represents how close we want the values of the function to be to the limit. It's a small positive number, and the closer we want to be, the smaller \( \epsilon \) becomes.
  • Delta (\( \delta \)): This indicates how close the input, or x-values, need to be to the point of interest, denoted as \( x_0 \). Once we decide on \( \epsilon \), we find this \( \delta \) to match it.
  • The goal is: If we can find a \( \delta \) for any chosen \( \epsilon \) so that for all x within the distance \( \delta \) (not equal to \( x_0 \)), the function value is within \( \epsilon \) of the limit, then the function \( f(x) \) approaches that limit as x approaches \( x_0 \).
This definition gives us the rigorous ground to claim a limit exists.
Direct substitution
Direct substitution is a straightforward approach to finding the limit of a function. If a function is easy and straightforward, we simply plug the point into the function. Let's explore what this means:
  • If a function is continuous at a point, direct substitution often allows you to find the limit instantly. This means you simply replace x with \( x_0 \) and evaluate the function.
  • For example, consider the function \( f(x) = 4 - 12x \). To find \( \lim_{x \rightarrow x_0} (4 - 12x) \), we substitute x with \( x_0 \) to get \( 4 - 12x_0 \). Here, the limit is already visible at \( L = 4 - 12x_0 \).
    • If the calculation after substitution results in a defined number, the limit exists and equals that number. If not, we may need the epsilon-delta method or algebraic manipulation to determine the limit.
Direct substitution works smoothly in cases of simple functions but be cautious of functions with indeterminacy or discontinuities.
Continuous functions
A function is continuous at a point if you can draw it without lifting your pen off the paper at that point. The great thing about continuous functions is how predictable they are when determining limits.
  • Mathematically, a function \( f(x) \) is continuous at \( x_0 \) if \( \lim_{x \rightarrow x_0} f(x) = f(x_0) \). This means that the function's limit as x approaches \( x_0 \) is simply the function's value at \( x_0 \).
  • In our example, \( f(x) = 4 - 12x \) is a linear function, which is continuous everywhere. Therefore, finding the limit at any point is straightforward: just substitute the point into the function.
  • Continuous functions do not break or jump, making the process of calculating limits, especially via direct substitution, efficient and intuitive.
When dealing with continuous functions, one can reliably use direct substitution to understand limits, as their straightforward behavior assures us that the limit is truly the value of the function at that point.

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