Problem 8
Question
$$ \lim _{x \rightarrow 1} \frac{\ln x^{2}}{x^{2}-1} $$
Step-by-Step Solution
Verified Answer
The limit is 1.
1Step 1: Recognize the Form
Substitute the value of the limit into the function to determine the form.\[\lim_{x \to 1} \frac{\ln x^2}{x^2 - 1} = \frac{\ln(1)^2}{1^2 - 1} = \frac{0}{0}\]This is an indeterminate form of \(\frac{0}{0}\), so we will need to use L'Hôpital's Rule.
2Step 2: Apply L'Hôpital's Rule
Since the limit results in \(\frac{0}{0}\), apply L'Hôpital's Rule, which involves differentiating the numerator and denominator separately.Differentiate the numerator: \[\frac{d}{dx}(\ln x^2) = \frac{d}{dx}(2\ln x) = \frac{2}{x}\]Differentiate the denominator: \[\frac{d}{dx}(x^2 - 1) = 2x\]The limit now becomes:\[\lim_{x \to 1} \frac{\frac{2}{x}}{2x}\]
3Step 3: Simplify and Evaluate the Limit
Simplify the expression obtained from L'Hôpital's Rule.\[\lim_{x \to 1} \frac{\frac{2}{x}}{2x} = \lim_{x \to 1} \frac{2}{2x^2} = \lim_{x \to 1} \frac{1}{x^2}\]Substitute \(x = 1\):\[\frac{1}{1^2} = 1\]Thus, the limit evaluates to 1.
Key Concepts
L'Hôpital's RuleIndeterminate FormsDifferentiationNatural Logarithm
L'Hôpital's Rule
Calculating limits in calculus can sometimes lead to an indeterminate form, such as \(\frac{0}{0}\). This is where L'Hôpital's Rule comes in handy. L'Hôpital's Rule provides a method to evaluate these limits by differentiating the numerator and denominator separately. This simplifies the complex limit into a more straightforward calculation. Remember, you can only use L'Hôpital's Rule when your direct substitution into the limit results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). After differentiation, if the result still leads to an indeterminate form, you may apply the rule again until the limit becomes determinate.
Underlying L'Hôpital's Rule is the idea that if two functions are approaching zero or infinity at a point, the behavior of their ratios is determined by the behavior of the ratios of their derivatives.
Underlying L'Hôpital's Rule is the idea that if two functions are approaching zero or infinity at a point, the behavior of their ratios is determined by the behavior of the ratios of their derivatives.
Indeterminate Forms
Limits can sometimes produce forms like \(\frac{0}{0}\), which are referred to as indeterminate forms. These forms don't yield a clear outcome directly, requiring further analysis with additional methods, such as L'Hôpital's Rule. Recognizing an indeterminate form is crucial.
These forms occur when both the numerator and the denominator of a fraction tend towards zero or both tend towards infinity. In many cases, they indicate that the direct substitution in the limit provides no clear value, hence indicating other methods like L'Hôpital’s Rule, algebraic rearrangement, or series expansion must be used.
These forms occur when both the numerator and the denominator of a fraction tend towards zero or both tend towards infinity. In many cases, they indicate that the direct substitution in the limit provides no clear value, hence indicating other methods like L'Hôpital’s Rule, algebraic rearrangement, or series expansion must be used.
- Common indeterminate forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0\cdot\infty\), \(\infty - \infty\), \(1^{\infty}\), \(0^0\), and \(\infty^0\).
- Addressing indeterminate forms usually involves simplifying the expression or finding equivalent expressions that are determinate.
Differentiation
Differentiation is a core concept in calculus used to find the rate at which a function changes at any given point. It involves finding the derivative of a function, which provides the slope of the tangent line to the curve at a particular point. In L'Hôpital's Rule, differentiation of the numerator and denominator turns a complex indeterminate form into an easier limit to solve.
The process of differentiation involves applying certain rules, such as the power rule, product rule, and chain rule, to systematically find the derivative. In our exercise, differentiating \(\ln(x^2)\) used a chain rule approach:
The process of differentiation involves applying certain rules, such as the power rule, product rule, and chain rule, to systematically find the derivative. In our exercise, differentiating \(\ln(x^2)\) used a chain rule approach:
- The "outside function" \(\ln(x^2)\) becomes \(2\ln x\).
- Applying the chain rule then gives the derivative \(\frac{2}{x}\).
Natural Logarithm
The natural logarithm, represented as \(\ln x\), is a logarithm with the base \(e\), where \(e\) is approximately equal to 2.71828. It is a unique logarithm due to its natural properties which make it appear frequently in calculus and natural growth problems.
Understanding its properties is essential as it helps in simplifying and differentiating expressions. The derivative of \(\ln x\) is \(\frac{1}{x}\), which is vital in solving calculus problems involving differentiation and L'Hôpital’s Rule.
Understanding its properties is essential as it helps in simplifying and differentiating expressions. The derivative of \(\ln x\) is \(\frac{1}{x}\), which is vital in solving calculus problems involving differentiation and L'Hôpital’s Rule.
- The natural logarithm is the inverse of the exponential function \(e^x\).
- It expresses time it takes for compounded growth to reach a certain level, important in continuous growth equations.
- In our problem, using \(\ln x^2\) allows the simplification via properties like \(\ln a^b = b \ln a\).
Other exercises in this chapter
Problem 7
Evaluate each improper integral or show that it diverges. \(\int_{1}^{\infty} \frac{d x}{x^{1.00001}}\)
View solution Problem 7
Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow \infty} \frac{\ln \left(\ln x^{1000}\right)}{\ln x}$$
View solution Problem 8
Evaluate each improper integral or show that it diverges. \(\int_{10}^{\infty} \frac{x}{1+x^{2}} d x\)
View solution Problem 8
Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$\lim _{x \rightarrow(1 / 2)^{-}} \frac{\ln (4-8 x)^{2}}{\tan \pi x}$$
View solution