Problem 44 - Contemporary Precalculus

Question

Find the limit. $$\lim _{x \rightarrow-\infty} \frac{|x|}{|x|+1}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches negative infinity is 0.

Step 1: Understand the absolute value

In this step, we need to understand how the absolute value function works. The absolute value of a number is always positive or zero. So, if x is a positive number, then |x| = x. If x is a negative number, then |x| = -x. For the given limit, since x approaches negative infinity, we know that x is a negative number. In this case, |x| = -x.

Step 2: Substitute the absolute value

Now we can substitute the absolute value of x with -x and rewrite the function as: $$\frac{-x}{-x+1}$$

Step 3: Simplify the function

To simplify the function, we can multiply the numerator and the denominator by (-1) to get rid of the negative signs: $$\frac{-x(-1)}{(-x+1)(-1)}$$ Which gives us: $$\frac{x}{x-1}$$

Step 4: Finding the limit

Now, finding the limit is simple. As x approaches negative infinity, the denominator x - 1 will also approach negative infinity making the entire fraction approaching zero. Therefore, the limit of the function is: $$\lim _{x \rightarrow-\infty} \frac{x}{x-1} = 0$$

Key Concepts

Absolute Value in Calculus

Absolute value is a way to express the distance of a number from zero on the number line, always as a non-negative value. Understanding absolute value is crucial in calculus, especially when dealing with limits involving negative or positive boundaries.

The expression \(|x|\) represents the absolute value of \(x\).

• If \(x\) is positive, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).

In the context of the given exercise, since \(x\) approaches negative infinity, \(x\) is a negative number. Thus, \(|x|\) effectively becomes \(-x\). This substitution is key to transforming the initial limit problem into a more straightforward expression. It's essential to keep in mind that the absolute value affects how we perceive expressions and needs special attention in calculations.

Simplifying Expressions with Substitutions

Simplifying algebraic expressions is an art in calculus that often involves substitution of complex parts with simpler forms. The ultimate goal is to make the problem easier to solve.

In this exercise, after acknowledging that \(|x| = -x\) for negative \(x\), the original expression \(rac{|x|}{|x|+1}\) becomes \(rac{-x}{-x+1}\).

• The next crucial step is to eliminate the negative signs for easier manipulation.
• This is done by multiplying both the numerator and the denominator by \(-1\), resulting in \(rac{x}{x-1}\).

This simplification gives us a clearer path to evaluate the limit. It's vital to remember that simplifying expressions often reveals the underlying behavior of functions, especially as they approach infinity or other boundaries. Simplification is not just about reducing terms; it’s about reinterpreting the mathematical expressions into a form we can effectively use.

Infinity in Calculus

Infinity is a concept that we encounter frequently in calculus. It's crucial to understand how functions behave as their inputs grow indefinitely large or small. This understanding enables us to define limits that provide valuable insights into the nature of the functions and equations.

In the problem given, we're exploring what happens to the expression \(rac{x}{x-1}\) as \(x\) tends to negative infinity.

• As \(x\) grows more negative, both \(x\) and \(x-1\) become very large in magnitude and negative.
• This results in the fractions’ behavior changing, resembling that of the function \(y = 1\), meaning it approaches zero.
• The approach to zero is because \(x-1\) becomes almost equal to \(x\) as both approach negative infinity.

Conclusively understanding how infinity affects expression and function behavior is critical when solving limits. It provides a framework for exploring the end-behavior of functions and understanding their practical implications in calculus.