Problem 50 - Calculus for Biology and Medicine

Question

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{u \rightarrow 2} \frac{4-u^{2}}{2-u} $$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Plug in the limit value

Begin by substituting the limit value into the expression. Substitute \( u = 2 \) into \( \frac{4 - u^2}{2 - u} \). This gives: \( \frac{4 - 2^2}{2 - 2} = \frac{4 - 4}{0} = \frac{0}{0} \). This indeterminate form requires further simplification.

Step 2: Simplify the expression using algebra

We need to simplify the expression \( \frac{4 - u^2}{2 - u} \). Notice that \( 4 - u^2 \) can be factored as a difference of squares: \((2 - u)(2 + u) \). Thus, the expression becomes \( \frac{(2 - u)(2 + u)}{2 - u} \).

Step 3: Cancel common factors

Cancel the common factor \( (2 - u) \) from the numerator and the denominator, remembering that \( u eq 2 \) to avoid division by zero. The expression simplifies to \( 2 + u \).

Step 4: Evaluate the limit of the simplified expression

Now substitute \( u = 2 \) into the simplified expression \( 2 + u \). This results in \( 2 + 2 = 4 \). Thus, \( \lim _{u \rightarrow 2} \frac{4-u^{2}}{2-u} = 4 \).

Key Concepts

Evaluating Limits

Limits are foundational in calculus, providing a way to understand behaviors of functions at specific points or as inputs become increasingly large or small. To evaluate a limit, such as \( \lim _{u \rightarrow 2} \frac{4-u^{2}}{2-u} \), begin by substituting the value into the expression. If the direct substitution yields a determinate form (like a non-zero number), it allows for straightforward computation. However, when substitution results in an indeterminate form (like \( \frac{0}{0} \)), we must explore further techniques like algebraic manipulation to find the limit. Evaluating limits, thus, often requires a strategic approach using different mathematical tools to get a meaningful answer.

Indeterminate Forms

Indeterminate forms are expressions where the limit cannot be directly determined through substitution because they are ambiguous. These forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), and \( \infty - \infty \), among others. In our exercise, substituting \( u = 2 \) directly into \( \frac{4-u^{2}}{2-u} \) resulted in the indeterminate form \( \frac{0}{0} \), signaling that basic substitution is insufficient. To resolve indeterminate forms, algebraic techniques, L'Hôpital's Rule, or other methods must be used to simplify and interpret the limit expression correctly.

Algebraic Simplification

Algebraic simplification plays a crucial role in resolving limits, especially when direct substitution leads to indeterminate forms. In the given problem, \( \frac{4-u^{2}}{2-u} \) was tackled using the difference of squares, simplifying it to \( \frac{(2 - u)(2 + u)}{2 - u} \). By recognizing common factors, you can cancel them out (as long as you ensure \( u eq 2 \) to avoid division by zero), reducing the expression to a simpler form \( 2 + u \). This simplification eliminates the indeterminate form, allowing for the direct substitution of \( u = 2 \) to yield the correct limit. Mastery of algebraic techniques is essential for solving complex limits effectively.