Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the following. Substitution may be helpful; these problems are variations on the theme \(\lim _{n \rightarrow \infty}\left(1+\frac{r}{n}\right)^{n}\). (a) \(\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}\) (b) \(\lim _{w \rightarrow \infty}\left(\frac{w+2}{w}\right)^{w}\) (c) \(\lim _{x \rightarrow \infty}\left(\frac{x-1}{x}\right)^{2 n}\) (d) \(\lim _{n \rightarrow \infty}\left(\frac{n}{n+1}\right)^{n}\) (e) \(\lim _{x \rightarrow 0^{+}}(1+2 x)^{3 /(2 x)}\)

Step-by-Step Solution

Answer

\n(a) The limit equals \( e \) \n(b) The limit equals \( e^2 \) \n(c) The limit equals \( 0 \) \n(d) The limit equals \( 1/e \) \n(e) The limit equals \( e^{3/4} \)

Step 1: Solve (a)

Rewrite the limit expression as \( \lim _{x \rightarrow 0^{+}}(1+x)^{1 /(1x)} \), which fits into the standard formula. Thus, the limit is \( e^1 = e \).

Step 2: Solve (b)

Rewrite the limit expression as \( \lim _{w \rightarrow \infty}\left(1+\frac{2}{w}\right)^{w} \). This also fits into the standard formula, so the limit is \( e^2 \).

Step 3: Solve (c)

The exponent in this limit expression doesn't match the formula's \(n\). However, we can plug in an arbitrary large number into this expression, and it will approach zero. Thus this limit is \(0^2n = 0\).

Step 4: Solve (d)

Rewrite this limit expression as \( \lim _{n \rightarrow \infty}\left(1+\frac{-1}{n+1}\right)^{n+1} \), which almost fits the standard formula, missing only a slight adjustment of the exponent which includes the denominator of the fraction inside the parenthesis, we instead use the exponent \( n+1 \). When the limit \( n \rightarrow \infty \), the terms \( n+1 \) and \( n \) approach the same infinite quantity and thus are interchangeable. This limit equals \( e^{-1} = 1/e \).

Step 5: Solve (e)

Rewrite the limit expression as \( \lim _{x \rightarrow 0^{+}}(1+2x)^{(3/2)/ (2x)} = \lim _{x \rightarrow 0^{+}}(1+2x)^{3 / (4x)} \). This fits into the formula and the limit is \( e^{3/4} \).

Key Concepts

Exponential Limits

Exponential limits often involve expressions raised to a power that tends to infinity or zero. A common problem type is identifying when an expression fits the form \(\left(1+\frac{r}{n}\right)^{n}\), which approaches the mathematical constant \(e\) as \(n\) becomes very large. For example, consider \(\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}\). When rewritten, it becomes \(\left(1+\frac{1}{n}\right)^{n}\), where \(n = \frac{1}{x}\), which is a standard exponential limit that simplifies to \(e\). Recognizing expressions that can be manipulated into this form is key. It allows us to evaluate complex expressions and apply limits to solve them efficiently.

Infinite Limits

Infinite limits are often approached in scenarios where a variable tends towards positive or negative infinity. These limits can simplify or converge into a more manageable form by evaluating specific expressions. An example is \(\lim _{w \rightarrow \infty}\left(\frac{w+2}{w}\right)^{w}\). By rewriting it as \(\left(1+\frac{2}{w}\right)^{w}\), it mirrors the classic exponential formula, satisfying the conditions as \(w \rightarrow \infty\). Thus, the limit takes the known value of \(e^{2}\). This transformation is crucial as it breaks down complex expressions into simpler forms by comparing them to familiar limit behaviors.

Evaluating Limits

Evaluating limits is an essential skill in calculus that involves substituting or rewriting expressions to find the limit of a function as it approaches a particular value. Key strategies include recognizing standard forms and using substitutions to simplify calculations. For instance, to solve \(\lim _{n \rightarrow \infty}\left(\frac{n}{n+1}\right)^{n}\), we rewrite it as \(\left(1-\frac{1}{n+1}\right)^{n+1}\), as this resembles the exponential limit form. When simplified, it equals \(e^{-1}\). Recognizing the potential for such transformations helps in systematically addressing seemingly complex problems and obtaining clear solutions.

Limit Substitution

Limit substitution is a technique often used when evaluating limits that do not initially seem to fit a standard form. By rewriting and simplifying expressions, we can match them to known limit results. Consider \(\lim _{x \rightarrow 0^{+}}(1+2x)^{3 /(2 x)}\). By breaking it down to \(\left(1+2x\right)^{3 / (4x)}\), we manipulate it into a suitable exponential form. This method allows us to efficiently compute the limit by equating it to familiar expressions, here resulting in \(e^{3/4}\). This substitution is vital for moving past initial complexities and arriving at straightforward solutions.