Problem 39
Question
Find the limit. $$ \lim _{x \rightarrow 1} \frac{\frac{1}{x+4}-\frac{1}{4}}{x} $$
Step-by-Step Solution
Verified Answer
\(\frac{-1}{20}\)
1Step 1: Simplify the complex fraction
Rewrite the fraction in simpler format, using the formula for the difference of two fractions which states \(\frac{a}{c} - \frac{b}{d} = \frac{ad - bc}{cd}\). So, the limit expression becomes \(\lim _{x \rightarrow 1} \frac{\frac{1}{x+4}-\frac{1}{4}}{x} = \lim _{x \rightarrow 1} \frac{\frac{4 - (x + 4)}{4(x+4)}}{x} = \lim _{x \rightarrow 1} \frac{-x}{4x(x+4)}\).
2Step 2: Cancel out common factor
Cancel out the common factor \(x\) in the numerator and denominator area, yielding \(\lim _{x \rightarrow 1} \frac{-1}{4(x+4)}\).
3Step 3: Substitute x with 1
Substitute \(x\) with 1 in the resulting expression to get the limit: \(\lim _{x \rightarrow 1} \frac{-1}{4(x+4)} = \frac{-1}{4(1+4)} = \frac{-1}{20}\).
Key Concepts
Complex FractionsLimits at a PointSimplifying Expressions
Complex Fractions
Complex fractions can seem confusing at first because they contain a fraction within a fraction. Imagine them like a fraction sandwich: a tiny fraction nested inside a larger one. To handle this layered structure, the first step is to simplify the internal fraction by finding a common denominator.
For example, in the exercise, you have \( \frac{1}{x+4} - \frac{1}{4} \). Here both terms have different denominators. You get a common denominator by multiplying these two denominators together—here, that is \(4(x+4)\). Now, rework the fractions:
For example, in the exercise, you have \( \frac{1}{x+4} - \frac{1}{4} \). Here both terms have different denominators. You get a common denominator by multiplying these two denominators together—here, that is \(4(x+4)\). Now, rework the fractions:
- Multiply the first term, \(\frac{1}{x+4}\), by \(4\) to get \(\frac{4}{4(x+4)}\).
- Multiply the second term, \(\frac{1}{4}\), by \((x+4)\) to get \(\frac{x+4}{4(x+4)}\).
Limits at a Point
Understanding limits at a point is vital because they help us determine the behavior of a function as it comes close to a particular value. Picture a limit like following a trail leading to the edge of a cliff where you never actually step over the edge. Instead, you get infinitely close.
In the exercise, we find the limit as \(x\) approaches 1. Practically, you substitute 1 where you see \(x\) after simplifying everything else. This substitution works because it helps to avoid division by zero errors that might occur without simplification. When the fraction was initially complex, substituting directly could have given undefined results.
By substituting afterward, as seen in the solution: \(\lim _{x \rightarrow 1} \frac{-1}{4(x+4)}\), it's straightforward. Put \(x = 1\) into the equation to find the actual limit value. The outcome \(\frac{-1}{20}\) confirms the limit exists and is finite.
In the exercise, we find the limit as \(x\) approaches 1. Practically, you substitute 1 where you see \(x\) after simplifying everything else. This substitution works because it helps to avoid division by zero errors that might occur without simplification. When the fraction was initially complex, substituting directly could have given undefined results.
By substituting afterward, as seen in the solution: \(\lim _{x \rightarrow 1} \frac{-1}{4(x+4)}\), it's straightforward. Put \(x = 1\) into the equation to find the actual limit value. The outcome \(\frac{-1}{20}\) confirms the limit exists and is finite.
Simplifying Expressions
Simplifying expressions is a critical step that makes complex math manageable. The aim is to reduce expressions to their simplest form to work with them easily. When you simplify, you might use distributive properties, factorization, cancellation, and substitution.
In the exercise, simplifying starts right after rewriting the fraction: \( \frac{4 - (x + 4)}{4(x+4)} \). Removing the brackets gives \(-x/4(x+4)\). By canceling out the common \(x\), the expression becomes \(\frac{-1}{4(x+4)}\). This process shows how to cancel terms precisely without changing the expression's value. By breaking it down slowly and carefully, you minimize errors.
A vital reminder: always perform cancellation on factors, not terms. Factors are multiplied parts of an expression, while terms are added or subtracted parts. In practice, this discipline ensures your simplification doesn't mistakenly invalidate or "break" the fractions you're working with.
In the exercise, simplifying starts right after rewriting the fraction: \( \frac{4 - (x + 4)}{4(x+4)} \). Removing the brackets gives \(-x/4(x+4)\). By canceling out the common \(x\), the expression becomes \(\frac{-1}{4(x+4)}\). This process shows how to cancel terms precisely without changing the expression's value. By breaking it down slowly and carefully, you minimize errors.
A vital reminder: always perform cancellation on factors, not terms. Factors are multiplied parts of an expression, while terms are added or subtracted parts. In practice, this discipline ensures your simplification doesn't mistakenly invalidate or "break" the fractions you're working with.
Other exercises in this chapter
Problem 39
Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{x^{2}-16}{x-4}\)
View solution Problem 39
Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility
View solution Problem 40
Use the General Power Rule to find the derivative of the function. $$ f(x)=(4-3 x)^{-5 / 2} $$
View solution Problem 40
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\left(3 x^{3}+4 x\right)(x-5)(x+1) $$
View solution