Problem 33 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Substitute the Limit Point into the Function

Since the function is continuous at the point we are considering, we can directly substitute the value of the limit into the function to find the limit. Thus, we evaluate the expression by substituting \( x = -1 \) into \( \sqrt{4 + 5x^4} \).

Step 2: Simplify the Expression Inside the Square Root

Calculate \( x^4 \) when \( x = -1 \). Since \((-1)^4 = 1\), we have:\[ 4 + 5(-1)^4 = 4 + 5 \times 1 = 9 \].

Step 3: Calculate the Square Root

Now, take the square root of the simplified expression: \[ \sqrt{9} = 3 \].

Key Concepts

Continuous functions

A continuous function is one where small changes in the input result in small changes in the output. In simpler terms, the graph of a continuous function is unbroken. It doesn't have any holes, jumps, or gaps. When dealing with limits in calculus, this property of continuous functions allows us to evaluate limits by simple substitution. This simply means that if a function is continuous at a point, you can find the limit by directly substituting the value of interest into the function.

• For example, consider the function \( f(x) = \sqrt{4 + 5x^4} \).
• This function is continuous for all real values of \( x \) since the expression under the square root is always a non-negative number, which means the square root is defined for those values.

Thus, at \( x = -1 \), we can directly find the limit by plugging in this value into the function without any extra steps. This simplicity is one of the powerful aspects of dealing with continuous functions in calculus.

Substitution method

The substitution method in calculus is an essential tool for finding limits, especially when dealing with continuous functions. As mentioned earlier, for functions that are continuous at a given point, the limit as \( x \) approaches that point can be found by directly substituting the value into the function.
Using substitution, you eliminate the need for other complex algebraic manipulations, especially with straightforward functions like polynomials or simple radicals. Here's how it applies to our exercise:

• To evaluate \( \lim _{x \rightarrow-1} \sqrt{4+5x^{4}} \), we use substitution because the function is continuous at \( x = -1 \).
• Simply replace \( x \, \text{with} \, -1 \) in the function to get \( \sqrt{4 + 5(-1)^4} \).
• This evaluates to \( \sqrt{9} \), which simplifies to 3.

This approach greatly simplifies the process of finding limits and helps in quickly obtaining the desired result without unnecessarily complicating the calculation process.

Square root simplification

Square root simplification is a critical concept, particularly when working with expressions under a root. When evaluating limits involving square roots, it's often necessary to simplify the expression inside the square root first.
In the given exercise, we have the expression \( \sqrt{4 + 5x^4} \). By substituting \( x = -1 \), we need to evaluate the inner expression first:

• Calculate \((-1)^4\), which equals 1.
• Substitute this into the expression: \(4 + 5 \cdot 1 = 9\).

Now that the expression under the square root is zeroed in on 9, we simplify by calculating the square root of 9, which is a straightforward arithmetic operation resulting in 3.
Square root simplification ensures precision and clarity in your work, especially when dealing with square roots in calculus, allowing you to achieve accurate results efficiently.