Question

find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$ \lim _{x \rightarrow \pi} \frac{2 x^{2}-6 x \pi+4 \pi^{2}}{x^{2}-\pi^{2}} $$

Short answer

The limit is 0.

Step-by-step solution

Identify the Type of Limit

In the given limit \( \lim_{x \rightarrow \pi} \frac{2x^2 - 6x\pi + 4\pi^2}{x^2 - \pi^2} \), observe that the limit appears to be of an indeterminate form \( \frac{0}{0} \). This suggests that factoring or simplifying may be necessary.

Simplify the Denominator

Notice that the denominator \( x^2 - \pi^2 \) is a difference of squares. Factor it as \( (x - \pi)(x + \pi) \).

Simplify the Numerator

Rewrite the numerator \( 2x^2 - 6x\pi + 4\pi^2 \) by looking for factors that might cancel with the denominator. Note that this resembles a quadratic expression, which might be factored using known identities or simplification.

Factor the Numerator

Attempt to factor the quadratic polynomial in the numerator if possible. Though factorization may not be directly obvious, apply algebraic identities to recognize it effectively: \( 2x^2 - 6x\pi + 4\pi^2 = 2(x^2 - 3x\pi + 2\pi^2) \). Attempt factorization: observe it as \( 2(x-\pi)^2 \).

Simplify the Expression

Combine both factored forms for simplification. The expression now becomes \( \frac{2(x-\pi)^2}{(x-\pi)(x+\pi)} \). Cancel the common factor \( x-\pi \) from numerator and denominator.

Evaluate the Limit

After canceling the common factor, we are left with \( \frac{2(x-\pi)}{x+\pi} \). Substitute \( x = \pi \): \( \frac{2(\pi-\pi)}{\pi+\pi} = \frac{0}{2\pi} = 0 \). Thus, the limit is \( 0 \).

Key concepts

Indeterminate Forms

When evaluating limits, you might come across expressions that result in an indeterminate form. An indeterminate form like \( \frac{0}{0} \) occurs when both the numerator and the denominator tend to zero as \( x \) approaches a certain value. These forms are challenging because algebraic expressions that seem undefined can often be simplified or transformed to reveal meaningful results.

The importance of recognizing indeterminate forms lies in the fact that they indicate the need for further algebraic manipulation, such as factoring or simplifying the expression. Once simplified, these expressions can reveal finite limits, even when initially they seem troublesome.

• Identify indeterminate forms: Typically, \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \(0\cdot \infty\).
• Apply algebraic techniques: Simplification, factoring, or using known limits.
• Aim for clearer expressions: Reduce expressions to solvable forms to evaluate limits correctly.

Factoring Algebraic Expressions

Factoring is an essential algebraic tool in calculus, particularly when dealing with limits. It involves breaking down a complex expression into simpler factors or components. When evaluating limits, especially those resulting in indeterminate forms, factoring can often unveil hidden simplifications.

Consider the original problem: the denominator \( x^2 - \pi^2 \) is a straightforward difference of squares. This can be factored into \((x - \pi)(x + \pi)\), revealing potential cancellations. For the numerator, while it might initially seem complex, recognizing it as a quadratic polynomial and factoring it into \(2(x - \pi)^2\) simplifies the expression greatly.

• Identify factoring opportunities: Look for squares, cubes, or common terms.
• Apply known identities: Such as difference of squares, to simplify expressions.
• Use factoring to cancel terms: Simplifying expressions prior to limit evaluation.

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. They are quite common in calculus and often appear in limit problems. The expression in the given problem \( \frac{2x^2 - 6x\pi + 4\pi^2}{x^2 - \pi^2} \) is a perfect example of a rational function.

Evaluating limits of rational functions often involves addressing indeterminate forms. Simplifying these expressions through factoring can reveal hidden constants or lead to resolutions of undefined forms. It's important to understand how to manipulate these functions algebraically, revealing solutions that might not be apparent at first glance.

• Understand structure: Recognize numerator and denominator polynomials.
• Apply algebraic manipulation: Use factoring to simplify complex structures.
• Resolve indeterminate forms: Through simplification or division to find limits.

Quadratic Polynomials

Quadratic polynomials are algebraic expressions of the form \( ax^2 + bx + c \). In rational functions or limit problems, they require careful attention for simplification or factoring. In the given problem, the numerator \( 2x^2 - 6x\pi + 4\pi^2 \) is a quadratic polynomial that we effectively simplified by recognizing it as \( 2(x - \pi)^2 \).

This kind of algebraic manipulation turns initially complex expressions into manageable forms, making the limit evaluation process smoother and more accurate. Identifying patterns such as perfect squares or factoring into binomials can greatly aid in these solutions.

• Identify quadratic forms: Recognize relations or patterns in coefficients.
• Use algebraic identities: Identify squares, trinomial factors, or binomial square patterns.
• Ensure simplification: Create opportunity for cancellation or substitution in limits.