Problem 49 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 4} \frac{\frac{1}{x}-\frac{1}{4}}{x-4}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches 4 is $\frac{-1}{16}$.

Step 1: Show the function's limit

Let's find the limit of the given function as x approaches 4. $$\lim _{x \rightarrow 4} \frac{\frac{1}{x}-\frac{1}{4}}{x-4}$$

Step 2: Simplify the complex fraction

The first step is to simplify the complex fraction. In order to do that, we find the least common denominator (LCD) of the two fractions in the numerator, which in this case is \(4x\). Multiply both fractions in the numerator of the complex fraction by the LCD, and simplify the expression. $$ \lim _{x \rightarrow 4} \frac{\frac{4 - x}{4x}}{x-4}$$

Step 3: Simplify the expression further by factoring

Now, we can factor the \(-1\) in the numerator, so that it will look simpler and be easier to work with. $$ \lim _{x \rightarrow 4} \frac{-1( x - 4)}{4x(x-4)}$$

Step 4: Cancel terms and evaluate the limit

Next, we notice that both the numerator and denominator have a \((x-4)\), so they cancel out. This leaves us with: $$\lim _{x \rightarrow 4} \frac{-1}{4x}$$ Now, we can substitute \(x = 4\) in the simplified expression to find the limit: $$\frac{-1}{4(4)} = \frac{-1}{16}$$ So, the limit of the given function is: $$\lim _{x \rightarrow 4} \frac{\frac{1}{x}-\frac{1}{4}}{x-4} = \frac{-1}{16}$$

Key Concepts

Limit of a Function

In calculus, the limit of a function is a fundamental concept that describes the behavior of the function as the input approaches a certain value. For example, when we write \( \lim_{x \rightarrow a} f(x) \) we are interested in finding out what value the function \( f(x) \) approaches as \( x \) gets closer and closer to \( a \).

In the given exercise, we're asked to find \( \lim _{x \rightarrow 4} \frac{\frac{1}{x}-\frac{1}{4}}{x-4} \). This limit is crucial in understanding the behavior of the function at the point \( x = 4 \), which is part of the concept of continuity. If a function is not continuous at a point, the limit at that point may not exist.

Simplifying Complex Fractions

Complex fractions have other fractions in either their numerator, their denominator, or both. Simplifying complex fractions involves multiple steps, including finding a common denominator, combining fractions, and reducing.

To simplify a complex fraction, one generally multiplies the numerator and the denominator by the least common denominator (LCD) to eliminate the smaller fractions. This process is evident in the solution where multiplying by the LCD of \( 4x \) allows the complex fraction to be expressed as a simpler one. Simplification is not just for appearances; it often reveals cancellations that make computing limits feasible.

LCD

The Least Common Denominator (LCD) is the smallest number that all denominators in a given set of fractions can divide into evenly. It's an essential step in adding, subtracting, and simplifying fractions.

When working with complex fractions, the LCD is used to combine fractions into a single fraction. In our limit example, the LCD of \( 1/x \) and \( 1/4 \) is \( 4x \) which helps to simplify the complex fraction. Understanding how to find the LCD is crucial in simplification processes.

Factoring Algebraic Expressions

Factoring is a mathematical process that involves breaking down an algebraic expression into simpler pieces, or factors, that when multiplied together yield the original expression. Factoring is a valuable tool since it can simplify algebraic expressions and solve equations more readily.

In limit problems, factoring can reveal terms that may cancel, as seen in the solution provided. After factoring \( -1 \) out, the term \( x - 4 \) appears in both the numerator and the denominator, which then cancels out to simplify further. This simplification step is crucial for correctly evaluating the limit of algebraic expressions.