Problem 125 - Advanced Calculus

Question

Use L'Hospital's Rule to evaluate (a) \(\lim _{x \rightarrow 2}\left\\{\left(2 x^{2}-4 x\right) /(x-2)\right\\}\) (b) \(\lim _{\mathrm{x} \rightarrow 0} \mathrm{x} \ln \mathrm{x}\) (c) \(\lim _{\mathrm{x} \rightarrow 0}(1 / \sin \mathrm{x})-(1 / \mathrm{x})\) (d) \(\lim _{\mathrm{x} \rightarrow \infty}(1+\mathrm{x})^{1 / \mathrm{x}}\)

Step-by-Step Solution

Answer

To summarize, the limits are: (a) \(\lim_{x \rightarrow 2}\left\{\left(2 x^{2}-4 x\right) /(x-2)\right\} = 4\) (b) \(\lim_{x \rightarrow 0} x \ln x = 0\) (c) \(\lim_{x \rightarrow 0}(1 / \sin x) - (1 / x) = 0\) (d) \(\lim_{x \rightarrow \infty}(1+x)^{1 / x} = 1\)

Step 1: Differentiate the numerator and denominator

Differentiate the numerator and the denominator separately with respect to x: Numerator: \(2x^2 - 4x \rightarrow 4x - 4\) Denominator: \(x - 2 \rightarrow 1\)

Step 2: Evaluate the limit

Now that we have the derivatives, we can evaluate the limit: \(\lim_{x \rightarrow 2} \frac{4x - 4}{1}= \frac{4(2) - 4}{1} = \boxed{4}\) (b)

Step 1: Rewrite the expression as a quotient

Rewrite the expression as a quotient of two functions: \(x\ln x = \frac{\ln x}{1/x}\)

Step 2: Differentiate the numerator and denominator

Differentiate the numerator and the denominator separately with respect to x: Numerator: \(\ln x \rightarrow 1/x\) Denominator: \(1/x \rightarrow -1/x^2\)

Step 3: Evaluate the limit

Now that we have the derivatives, we can evaluate the limit: \(\lim_{x \rightarrow 0} \frac{1/x}{-1/x^2}= \lim_{x \rightarrow 0} (-x) = \boxed{0}\) (c)

Step 1: Rewrite the expression as a quotient

Rewrite the expression as a quotient of two functions: \(\frac{1}{\sin x} - \frac{1}{x} = \frac{x-\sin x}{x \sin x}\)

Step 2: Differentiate the numerator and denominator

Differentiate the numerator and the denominator separately with respect to x: Numerator: \(x - \sin x \rightarrow 1 - \cos x\) Denominator: \(x\sin x \rightarrow \sin x + x\cos x\)

Step 3: Evaluate the limit

Now that we have the derivatives, we can evaluate the limit: \(\lim_{x \rightarrow 0} \frac{1-\cos x}{\sin x + x\cos x}=\frac{0}{0}\), Applying L'Hospital's rule one more time. Numerator: \(1-\cos x \rightarrow \sin x\) Denominator: \(\sin x + x\cos x \rightarrow \cos x -x\sin x\) Now, evaluating the limit: \(\lim_{x \rightarrow 0} \frac{\sin x}{\cos x - x\sin x}=\frac{0}{1} =\boxed{0}\) (d)

Step 1: Rewrite the expression using natural logarithm

Rewrite the expression using the natural logarithm and rewrite it as a quotient: \(\ln\left((1+x)^{1/x}\right)= \frac{\ln(1+x)}{x}\)

Step 2: Differentiate the numerator and denominator

Differentiate the numerator and the denominator separately with respect to x: Numerator: \(\ln(1+x) \rightarrow \frac{1}{1+x}\) Denominator: \(x \rightarrow 1\)

Step 3: Evaluate the limit

Now that we have the derivatives, we can evaluate the limit: \(\lim_{x \rightarrow \infty} \frac{\frac{1}{1+x}}{1}= \lim_{x \rightarrow \infty} \frac{1}{1+x}=0\) Now we need to rewrite the expression to compute the original limit: \(\lim_{x \rightarrow \infty}(1+x)^{1 / x}= e^{\lim_{x\to\infty}\frac{\ln(1+x)}x} = e^0 = \boxed{1}\)

Key Concepts

Limits in Calculus

Understanding Limits in Calculus is crucial as they form the foundation of more advanced concepts like differentiation and integration. A limit is essentially an expected value that a function approaches as the input variable gets closer to some specified point.

For example, when calculating the limit of \(f(x)\) as \(x\) approaches a certain value \(a\), denoted as \(\lim_{x \rightarrow a} f(x)\), we're trying to find out what value \(f(x)\) gets very close to - or reaches - when \(x\) is near \(a\). Limits allow us to handle situations where the function might not be directly defined at point \(a\), or the function approaches an infinite value.

Moreover, limits help in understanding the behavior of functions at points of discontinuity or where they have undefined values — known as indeterminate forms. They are essential in establishing the foundation for the concept of continuity, a key concept that indicates whether a function is well-behaved and predictable across its domain.

Differentiation

The process of Differentiation is a fundamental tool in calculus, representing the rate at which a function is changing at any given point. It's the mathematically rigorous way to describe and calculate the concept of an `instantaneous rate of change`, which can be thought of as the slope of the tangent line to a point on a curve.

When we differentiate a function \(f(x)\), we're looking for another function, called the derivative of \(f\), denoted as \(f'(x)\) or \(\frac{df}{dx}\). The derivative tells us how fast the value of \(f(x)\) is changing for any value of \(x\). In the context of textbook problems, differentiation allows us to transform complex limits that involve indeterminate forms into simpler ones that we can more easily evaluate, as seen in the demonstration of L'Hospital's Rule.

Knowing how to differentiate various functions - from simple polynomials to more complex trigonometric and logarithmic functions - is crucial. Each type of function has its set of rules to follow for differentiation, such as the power rule, product rule, chain rule, and more, which should be mastered for effective application in calculus problems.

Indeterminate Forms

In calculus, an Indeterminate Form is an expression we encounter in limit problems that does not immediately reveal information about the actual limit of a function as inputs approach a particular point. Examples include \(0/0\), \(\infty/\infty\), \(0 \times \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\).

An indeterminate form is not simply undefined; rather, it suggests that the limit could exist but requires further analysis to discover. This is where L'Hospital's Rule becomes exceptionally helpful. It states that if the limit of a function leads to an indeterminate form of type \(0/0\) or \(\infty/\infty\), then the limit can be computed by differentiating the numerator and denominator separately and taking the limit of the resulting fraction.

This rule transforms an indeterminate form into a determinate one, allowing us to evaluate the limit without ambiguity. Despite its power, L'Hospital's Rule can only be applied under certain conditions and may require multiple applications if the initial differentiation does not resolve the indeterminate form. Practicing with a variety of limits that result in indeterminate forms is essential for grasping the nuances of this technique and its proper application in different scenarios.