Problem 37 - Calculus of a Single Variable

Question

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow-3} \frac{x+3}{x^{2}+x-6} $$

Step-by-Step Solution

Answer

The one-sided limit of the given function as \(x\) approaches -3 from the left is -0.2.

Step 1: Factorize the denominator

The first step to simplify the function is factoring the denominator. So, Dennis topic can Factorize \(x^{2}+x-6\) into \((x-2)(x+3)\).

Step 2: Simplify the fraction

After factoring, you can simplify the fraction \(\frac{x+3}{(x-2)(x+3)}\) by canceling out the common terms from numerator and denominator, which will reduce the fraction to \(\frac{1}{x-2}\) for \(x\) not equals to -3.

Step 3: Compute the one-side limit

With the reduced expression \(\frac{1}{x-2}\), it's possible to find the limit as \(x\) approaches -3 from the left by substituting -3 in the place of \(x\) in the \(x-2\) denominator. That results in \(\frac{1}{-3-2}\) which simplifies to -0.2

Key Concepts

Factorization

Factorization is a method used to break down complex algebraic expressions into simpler components or factors. It can be particularly helpful for simplifying rational expressions. In our exercise, we are tasked with simplifying the denominator of the function \[ \frac{x+3}{x^{2}+x-6} \]The first step involves identifying and extracting the common factors from the quadratic expression in the denominator. This quadratic expression, \[ x^2 + x - 6, \]can be rewritten as a product of two simpler binomials. By factoring, we find that:\[ x^2 + x - 6 = (x - 2)(x + 3). \]For beginners, it often helps to look for two numbers that multiply to the constant term (-6) and add to the linear coefficient (1). The numbers that fit these criteria here are 3 and -2. This allows us to rewrite the expression in this neat factored form, which is essential for simplifying the limit problem.

Limit Calculation

Limit calculation is a crucial aspect of calculus that evaluates the behavior of a function as it approaches a specific point. In this problem, we are asked to find the limit of the given function \[ \lim _{x \rightarrow-3} \frac{x+3}{x^{2}+x-6} \]from the left side.After successfully factoring the denominator and cancelling out the common terms from the numerator and denominator, we simplify the expression to \[ \frac{1}{x-2} \] for \( x eq -3 \). This simplification allows us to focus directly on the behavior of the remaining term near the point of interest, \( x = -3 \).To compute the limit, substitute the approach value, -3, into the simplified expression:\[ \frac{1}{x - 2} \rightarrow \frac{1}{-3 - 2} = \frac{1}{-5}. \]This results in the limit evaluating to -0.2, illustrating how the function behaves as the input value moves closer to -3 from the left.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about reducing expressions to their most manageable form, making them easier to evaluate or analyze. This often involves factoring, canceling terms, and performing arithmetic operations simplistically. In our exercise, the expression\[ \frac{x+3}{x^2+x-6} \]manual simplification began by factoring the denominator:1. Identified common factors, \((x+3)\), between numerator and denominator.2. Canceled out identical terms to yield a simpler expression, \(\frac{1}{x-2}\). Simplification plays a vital role in not only solving but comprehending limits more deeply. Simplifying an equation involves transforming it into a form where the unknown value (here, a limit) is easily substitutable, thus making complicated functions easier to approach and conclude, just as we did leading to the solution.