Question

Evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 0} \frac{(1+x)^{n}-1-n x}{x^{2}} $$

Short answer

The limit is \( \frac{n(n-1)}{2} \).

Step-by-step solution

Identify the form of the limit

The limit given is \( \lim_{x \rightarrow 0} \frac{(1+x)^{n} - 1 - nx}{x^{2}} \). At \( x = 0 \), both the numerator and denominator become zero. This is an indeterminate form \( \frac{0}{0} \), which means we can use L'Hôpital's rule.

Apply L'Hôpital's Rule

To apply L'Hôpital's rule, we first differentiate the numerator and the denominator. The derivative of the denominator, \( x^2 \), is \( 2x \). The derivative of the numerator \( (1+x)^n - 1 - nx \) is more complex and requires separate differentiation.

Differentiate the numerator

Differentiate \( (1+x)^n \) using the chain rule. This gives \( n(1+x)^{n-1} \). We also have the terms \( -1 \) whose derivative is 0 and \( -nx \) whose derivative is \( -n \). Therefore, the derivative of the numerator is \( n(1+x)^{n-1} - n \).

Apply L'Hôpital's Rule Again or Simplify

After differentiating, the expression becomes: \( \lim_{x \to 0} \frac{n(1+x)^{n-1} - n}{2x} \). At \( x = 0 \), both numerator and denominator are 0 again. So, use L'Hôpital's Rule a second time by differentiating the numerator and denominator once more.

Differentiate the numerator a second time

Differentiate \( n(1+x)^{n-1} - n \) to get \( n(n-1)(1+x)^{n-2} \). Differentiate the denominator \( 2x \) to get 2. So, the expression becomes \( \lim_{x \rightarrow 0} \frac{n(n-1)(1+x)^{n-2}}{2} \).

Evaluate the limit directly

Substitute \( x = 0 \) into \( \frac{n(n-1)(1+x)^{n-2}}{2} \) to get \( \frac{n(n-1)}{2} \). Since \( (1+0)^{n-2} = 1 \), the numerator simplifies to \( n(n-1) \).

Write the final answer

Thus, the limit is \( \frac{n(n-1)}{2} \).

Key concepts

L'Hôpital's Rule

When evaluating limits in calculus, you may encounter situations where you are dealing with an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In such cases, L'Hôpital's Rule is a powerful tool that can simplify your calculations. This rule states that if the limits of both the numerator and the denominator of a given function tend to 0 or infinity, the limit of the original function can be evaluated by taking the limit of the derivatives of the numerator and denominator instead.

To apply L'Hôpital's Rule:

• First, verify that you have an indeterminate form (like \( \frac{0}{0} \)).
• Then, differentiate both the numerator and the denominator separately.
• Finally, evaluate the limit of the new fraction.

Sometimes you may need to apply the rule more than once to resolve the indeterminate form.

Indeterminate Forms

Indeterminate forms are expressions that do not have a well-defined limit at first glance. Common examples include \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). These forms arise during limit evaluation and suggest that direct substitution will not work as the limit cannot be determined easily.

There are several types of indeterminate forms, including:

• \( 0^0 \)
• \( 1^\infty \)
• \( \infty - \infty \)
• \( 0 \times \infty \)

Identifying an indeterminate form is crucial as it often guides the use of specific techniques such as L'Hôpital's Rule or algebraic manipulation, to resolve the limit.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that involves finding the value that a function approaches as the input approaches a specific point. In the exercise given, you'll learn how to evaluate limits that initially present with indeterminate forms by using L'Hôpital's Rule and algebra.

To effectively evaluate limits:

• Check if substitution of the point gives a direct answer. If not, identify any indeterminate forms.
• When confronted with a \( \frac{0}{0} \) form, consider applying L'Hôpital's Rule by differentiating both the numerator and the denominator.
• In cases where L'Hôpital's Rule continuously yields an indeterminate form, reconsider the expression or try simplifying further.

This strategy lets you circumvent computation complexities and find the desired limit. The goal is to transform the expression into a form where the limit can be directly calculated.

Differentiation Techniques

Differentiation is a vital technique in calculus that involves finding the derivative, or the rate of change, of a function. In the process of using L'Hôpital's Rule, differentiating the numerator and the denominator plays a key role.

Some common differentiation techniques include:

• Power Rule: For \( x^n \), the derivative is \( nx^{n-1} \).
• Chain Rule: Used for composite functions, such as \( (1+x)^n \), where the derivative is \( n(1+x)^{n-1} \).
• Product Rule: For functions that are a product of two functions, like \( u(x) \cdot v(x) \), differentiate using \( u'v + uv' \).
• Quotient Rule: Used for divisions such as \( \frac{u(x)}{v(x)} \).

By mastering these differentiation methods, you can quickly solve for the derivatives needed when applying L'Hôpital's Rule to evaluate limits.