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Chapter 1: Problem 25

Find the limits. $$ \lim _{x \rightarrow 4^{-}} \frac{3-x}{x^{2}-2 x-8} $$

Short Answer

Expert verified
The limit is \(+\infty\).

Step by step solution

01

Identify the expression

The given limit to solve is \( \lim_{x \to 4^{-}} \frac{3-x}{x^2 - 2x - 8} \). Our goal is to simplify this expression and find its limit as \( x \) approaches 4 from the left.
02

Factor the quadratic expression

The denominator \( x^2 - 2x - 8 \) can be factored. We need factors of \( -8 \) that add up to \( -2 \). These factors are \( +2 \) and \( -4 \). Therefore, \( x^2 - 2x - 8 = (x - 4)(x + 2) \).
03

Substitute and simplify

Substitute the factored form into the expression: \( \frac{3-x}{(x-4)(x+2)} \). It doesn't seem to simplify directly, but as \( x \to 4 \), evaluate the behavior of the expression instead.
04

Evaluate the limit approaching from the left

As \( x \to 4^{-} \), \( x - 4 \) approaches 0 from the negative side. The expression becomes undefined, but we can explore its behavior. Approaching from the left, \( 3-x \) approaches \( 3-4 = -1 \). Therefore, the numerator is negative. The denominator approaches \( 0 \times 6 \) which is zero, but \((x-4)\) is negative, so the whole fraction approaches positive infinity.

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Key Concepts

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

Limit Evaluation
When evaluating limits, the primary aim is to find a value that a function approaches as the variable gets closer to a specific point. In this exercise, we are asked to find the limit \( \lim_{x \to 4^{-}} \frac{3-x}{x^2 - 2x - 8} \).Limit evaluation involves a few critical steps:
  • First, identify any indeterminate forms that occur when substituting the limit point into the function.
  • Second, simplify expressions algebraically to help resolve these forms.
  • Finally, analyze the behavior of the function from either side of the limit point. Here, as \( x \) approaches 4 from the left, understanding whether the function increases or decreases is crucial.
It's important to note that if simplification cannot help eliminate indeterminacies, observing function behavior through graphing or algebraic manipulation can often provide insights into the limit. This ensures you find the limit correctly.
Quadratic Factorization
Quadratic factorization is a handy technique used to simplify expressions, especially when dealing with rational functions. Often, a quadratic can be factored into a product of two binomials. In the denominator of our function,\( x^2 - 2x - 8 \) needs to be factored.Factorizing quads follows these simple steps:
  • Identify the quadratic in standard form: \( ax^2 + bx + c \).
  • Look for two numbers that multiply to \( ac \) (product of the first and last coefficients) and add to \( b \) (middle coefficient).
  • These numbers in our case are \(+2\) and \(-4\), allowing us to rewrite the quadratic as \( (x - 4)(x + 2) \).
Once factorized, it's easier to either cancel terms with the numerator or understand the function's behavior near a point. This skill is essential as it frequently appears across calculus problems and is a prerequisite for simplifying limits.
Infinite Limits
Infinite limits typically occur when a function's value grows without bound as the input approaches a certain point. In our scenario, as \( x \to 4^{-} \), the function \( \frac{3-x}{(x-4)(x+2)} \) approaches positive infinity.Here鈥檚 how infinite limits can be comprehended:
  • As \( x \) nears 4 from the left (just below 4), \( x-4 \) becomes a small negative number causing division by nearly zero.
  • The numerator \( 3-x \) evaluates to \(-1\), indicating a negative top.
  • The denominator \((x-4)\) emphasizes the small negative, driving the fraction's magnitude to grow positively larger.
Thus, coping with infinite limits includes understanding asymptotic behavior where the fraction鈥檚 value skyrockets without settling, explaining why the limit is positive infinity in this context. Identifying these infinite responses helps illuminate more complex function behaviors in advanced calculus.

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