Problem 43 - Calculus for Biology and Medicine

Question

In Problems \(29-48\), find the limits. $$ \lim _{x \rightarrow-2} \frac{1}{\sqrt{5 x^{2}-4}} $$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{4}\).

Step 1: Recognizing the Limit Expression

The given exercise requires us to find the limit of the function as \(x\) approaches \(-2\). The expression is \(\lim _{x \rightarrow -2} \frac{1}{\sqrt{5x^2 - 4}}\). Our goal is to find the value that this expression approaches as \(x\) gets closer and closer to \(-2\).

Step 2: Substitute and Simplify

Substitute \(x = -2\) directly into the expression: \(\frac{1}{\sqrt{5(-2)^2 - 4}}\). This becomes \(\frac{1}{\sqrt{20 - 4}}\). Simplifying further, it is \(\frac{1}{\sqrt{16}}\).

Step 3: Compute the Final Value

Simplify \(\sqrt{16}\) which is 4. Thus, the expression becomes \(\frac{1}{4}\). So, the limit of the expression approaches \(\frac{1}{4}\) as \(x\) approaches \(-2\).

Key Concepts

Finding Limits

Finding limits is an essential concept in calculus that helps us understand what value a function approaches as its input gets closer to a particular point. Limits allow us to handle situations where direct evaluation might be tricky or impossible. To find a limit, we replace the variable in the expression with a value that approaches the point of interest and observe the behavior of the function near that point.

For instance, in the exercise given, we need to find the value of the expression \( \lim _{x \rightarrow -2} \frac{1}{\sqrt{5x^2 - 4}} \). The idea is to see what happens to the expression as \(x\) gets closer to \(-2\). You should remember that finding a limit successfully involves recognizing the expression's behavior and using algebraic manipulation or calculus techniques to simplify it.

Calculus Techniques

Calculus provides several techniques to find limits effectively. Some common ones include direct substitution, factoring, and rationalization, among others. Each technique has its purpose and is applicable in different situations.

In this example, direct substitution was employed. This means we directly substituted \(x = -2\) into the function. From there, we simplified the expression step by step.

• **Direct Substitution**: Plug the value directly whenever the expression is continuous and doesn't introduce division by zero or an undefined state.
• **Factoring/Rationalization**: Used when direct substitution doesn't work and can help to eliminate indeterminate forms.
• **Algebraic Manipulation**: Applies when adjustments or transformations make the limit visible or easier to evaluate.

Knowing which technique to use in each situation comes with experience, and practice will improve your understanding!

Simplifying Expressions

Simplifying expressions is a critical skill when dealing with limits, especially when the initial function looks complex. Simplification involves breaking down the expression into more manageable parts so that evaluating it becomes straightforward.

In our example, after substituting \(x = -2\), the expression became \(\frac{1}{\sqrt{20 - 4}}\), which is further simplified to \(\frac{1}{\sqrt{16}}\). Finally, this simplifies to \(\frac{1}{4}\). This process of simplification made it possible to easily determine the limit as \(x\) approaches \(-2\).

• Simplifying often involves combining like terms or reducing complex fractions.
• Recognizing squares, cubes, and common roots can make simplification faster and easier.
• The goal is always to make the expression easier to understand and evaluate.

By focusing on simplification, your path to finding the limit becomes clearer and less prone to error.