Problem 28 - APEX Calculus

Question

Evaluate the given limit. $$ \lim _{x \rightarrow \pi} \frac{3 x+1}{1-x} $$

Step-by-Step Solution

Answer

The limit is \( \frac{3\pi + 1}{1 - \pi} \).

Step 1: Plug in the Limit Value

First, substitute the value that \(x\) approaches into the expression: \( \lim _{x \rightarrow \pi} \frac{3x+1}{1-x} \). Replace \(x\) with \( \pi \).

Step 2: Substitute \( x = \pi \)

Substituting \(x = \pi \) into the function gives:\[ \frac{3(\pi) + 1}{1 - \pi} \].

Step 3: Simplify the Expression

Now calculate the numerator and the denominator separately:Numerator: \(3\pi + 1\).Denominator: \(1 - \pi\).Put them back into the fraction: \( \frac{3\pi + 1}{1 - \pi} \).

Step 4: Evaluate the Fraction

After substitution and simplification, the expression \( \frac{3\pi + 1}{1 - \pi} \) does not simplify further. Therefore, this is the limit as \(x\) approaches \(\pi\).

Key Concepts

Substitution Method

In limits, the substitution method is a straightforward way to approach finding the limit of a function. It involves directly replacing the variable with the value it approaches. This is especially useful when the function is continuous at the point of interest. In our exercise, where we have \( \lim _{x \rightarrow \pi} \frac{3x+1}{1-x} \), the first step is to see if substituting directly is possible and meaningful. The goal here is to simplify and evaluate if the expression remains determinate.

• Start by replacing \(x\) with \(\pi\) in the expression.
• If the substitution leads to a defined result (no division by zero or indeterminate forms such as \(\frac{0}{0}\)), we can conclude the limit calculation.

This method is incredibly handy for functions without particular behaviors, like discontinuities, at the point of interest. As it was applied to our problem, the substitution did yield a definite result, so we continued with simplifying it.

Simplifying Expressions

After substituting the value \(x = \pi\), simplification is the next logical step. Simplifying helps in cleaning up an expression so that any apparent complexity can be resolved. Here’s how we simplified \( \frac{3(\pi) + 1}{1 - \pi} \)

• Simplify the numerator: Calculate \(3\pi + 1\).
• Simplify the denominator: Calculate \(1 - \pi\).
• Reassemble these into a fraction that represents the simplest form.

This fraction \( \frac{3\pi + 1}{1 - \pi} \) doesn't reduce further, indicating that with this simplification, we have reached the simplest form of the expression.

Numerator and Denominator

Understanding a fraction's components is crucial in limit problems. The numerator and denominator each play a vital role in how we manipulate and understand the expression. For our problem, they formed \[3\pi + 1\] (numerator) and \[1 - \pi\] (denominator).

• The numerator tells us about how quickly the top part of the fraction grows or shrinks as \(x\) approaches the limit.
• The denominator impacts the overall behavior of the fraction, including if it might approach zero or become negative.

Recognizing these parts helps us anticipate whether any cancellation or further simplification might occur, potentially influencing the limit's final value. It's these elements that we pay attention to, especially if they involve a critical value like \(\pi\).

Limits in Calculus

The concept of limits forms the backbone of calculus, guiding our understanding of functions' behaviors near certain points. Limits establish how functions behave as inputs approach a particular value. For instance, \( \lim _{x \rightarrow \pi} \frac{3x+1}{1-x} \) helps describe the behavior of our function \( \frac{3x+1}{1-x} \) as \(x\) gets very close to \(\pi\).

• Limits help predict values where direct substitution isn't possible due to discontinuities or indeterminate forms.
• They are essential in defining derivatives and integrals, which rely on the behavior of functions as variables change infinitely.

In our example, the result after direct substitution and simplification gave a clear depiction of what the expression \( \frac{3x+1}{1-x} \) approaches as \(x\) nears \(\pi\). This shows the power and necessity of using limits to dissect a function's tendencies and nuances around specific points.