Problem 74
Question
Evaluate the limits using limit properties. If a limit does not exist, state why. $$\lim _{x \rightarrow 0} \frac{(x-1)^{3}+1}{x}$$
Step-by-Step Solution
Verified Answer
The limit is 3.
1Step 1: Identify the Type of Limit
We need to find \( \lim _{x \rightarrow 0} \frac{(x-1)^{3}+1}{x} \). The expression has the form \( \frac{0}{0} \) when directly substituting \( x = 0 \). This indicates an indeterminate form, so rearrangement or further manipulation is needed.
2Step 2: Simplify the Expression
The numerator \((x-1)^{3}+1\) can be expanded using the binomial theorem: \((x-1)^3 = x^3 - 3x^2 + 3x - 1\). Adding 1 to this gives \((x-1)^3 + 1 = x^3 - 3x^2 + 3x\). So the original expression becomes \(\frac{x^3 - 3x^2 + 3x}{x}\).
3Step 3: Factor Out Common Terms
In the expression \(\frac{x^3 - 3x^2 + 3x}{x}\), factor \(x\) out from the numerator. This results in \(\frac{x(x^2 - 3x + 3)}{x}\).
4Step 4: Cancel Out \(x\) from Numerator and Denominator
Cancel the \(x\) in the numerator and denominator. The limit now becomes \(\lim_{x \rightarrow 0} (x^2 - 3x + 3)\).
5Step 5: Evaluate the Limit
Substitute \( x = 0 \) into the simplified expression \(x^2 - 3x + 3\). This gives \(0^2 - 3 \times 0 + 3 = 3\).
6Step 6: Confirm the Limit Exists
Since we have simplified the expression and directly substituted the value of \(x\), and the result is finite, the limit exists and equals 3.
Key Concepts
Understanding Indeterminate FormsThe Binomial Theorem in ActionExploring Limit Properties
Understanding Indeterminate Forms
An indeterminate form is a situation in calculus where substituting the value for a variable in a limit expression results in an undefined value. Particularly, the fractions that result in forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) are classic scenarios of indeterminate forms. This occurs because both the numerator and denominator tend toward zero or infinity, making the expression undefined without further manipulation.
In the original problem, substituting \(x=0\) into the expression \(\frac{(x-1)^{3}+1}{x}\) results in \(\frac{0}{0}\). This tells us the form is indeterminate, necessitating further analysis or simplification. By recognizing this, you are prompted to either factor, rationalize, or use other algebraic methods to resolve the indeterminate nature of the limit. Understanding these forms is crucial because they require deeper investigation to find meaningful values for limits.
In the original problem, substituting \(x=0\) into the expression \(\frac{(x-1)^{3}+1}{x}\) results in \(\frac{0}{0}\). This tells us the form is indeterminate, necessitating further analysis or simplification. By recognizing this, you are prompted to either factor, rationalize, or use other algebraic methods to resolve the indeterminate nature of the limit. Understanding these forms is crucial because they require deeper investigation to find meaningful values for limits.
The Binomial Theorem in Action
The binomial theorem provides a method for expanding expressions that are raised to a power, specifically in the form \((a+b)^n\). For the exercise at hand, applying the binomial theorem simplifies the expression \((x-1)^3\).
Essentially, the theorem allows for the expansion as follows:
Understanding the binomial theorem not only simplifies complex polynomial expressions but is also essential for resolving indeterminate forms. By transforming the original expression, you make it easier to perform further algebraic manipulations, which then allow for limits like the one in the exercise to be evaluated.
Essentially, the theorem allows for the expansion as follows:
- \((x-1)^3 = x^3 - 3x^2 + 3x - 1\)
Understanding the binomial theorem not only simplifies complex polynomial expressions but is also essential for resolving indeterminate forms. By transforming the original expression, you make it easier to perform further algebraic manipulations, which then allow for limits like the one in the exercise to be evaluated.
Exploring Limit Properties
Limit properties are foundational rules that allow us to simplify the calculation of limits by performing algebraic operations directly. Once an expression is appropriately simplified, such properties include:
By substituting \(x=0\) into the simplified expression, we utilize the property of direct substitution, yielding the final limit. Understanding these properties enables tackling complex limit problems efficiently. When these properties are properly applied, they simplify the evaluation of limits, leading to quick and accurate determinations of whether limits exist and what their values might be.
- Direct substitution for continuous functions after simplifying the expression.
- Cancelation of common terms when valid.
- Splitting limits into simpler, individual fractions or terms.
By substituting \(x=0\) into the simplified expression, we utilize the property of direct substitution, yielding the final limit. Understanding these properties enables tackling complex limit problems efficiently. When these properties are properly applied, they simplify the evaluation of limits, leading to quick and accurate determinations of whether limits exist and what their values might be.
Other exercises in this chapter
Problem 73
Evaluate the limits using limit properties. If a limit does not exist, state why. $$\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$$
View solution Problem 73
Use Heron's formula to find the area of a triangle with sides \(a=5\) in. \(b=8\) in. \(c=9\) in. rounded to two decimal places.
View solution Problem 74
Write a sinusoidal equation given the maximum is \(100,\) the minimum is -20 and the period is 365
View solution Problem 75
Evaluate the limits using limit properties. If a limit does not exist, state why. $$\lim _{x \rightarrow 1} \frac{x^{3}-x^{2}}{\sqrt{x^{2}-2 x+1}}$$
View solution