Problem 31 - Calculus Early Transcendentals

Question

Use continuity to evaluate the limit. $$\lim _{x \rightarrow 4} \frac{5+\sqrt{x}}{\sqrt{5+x}}$$

Step-by-Step Solution

Answer

The limit is \( \frac{7}{3} \).

Step 1: Identify the Function

The given expression is \( f(x) = \frac{5+\sqrt{x}}{\sqrt{5+x}} \). We want to evaluate \( \lim_{x \to 4} f(x) \) by using continuity.

Step 2: Confirm Continuity

Both the numerator \( 5+\sqrt{x} \) and the denominator \( \sqrt{5+x} \) are continuous functions everywhere in their domain. The limit \( x \to 4 \) is within the domain, so we can evaluate the function directly at \( x=4 \).

Step 3: Substitute Directly into the Function

Since the function is continuous at \( x=4 \), we can directly substitute \( x=4 \) into the expression: \( f(4) = \frac{5+\sqrt{4}}{\sqrt{5+4}} \).

Step 4: Simplify the Expression

Simplify the substituted expression: \( f(4) = \frac{5+2}{\sqrt{9}} = \frac{7}{3} \).

Step 5: Conclude the Limit Evaluation

Since we obtained \( \frac{7}{3} \) from direct substitution in a continuous function, this is the value of the limit we were seeking.

Key Concepts

Continuity

Continuity is a key concept in calculus. It determines whether we can evaluate limits directly by substituting the point into the expression.
A function is considered continuous at a point if the limit as the function approaches the point equals the value of the function at that point. This means there are no sudden jumps, breaks, or holes at that specific point.
When asked to evaluate a limit using continuity, our first step is to check if the function is continuous at the given point. If it is continuous, we can confidently proceed to direct substitution.

Direct Substitution

Direct substitution is a straightforward method used to find the value of a limit for a continuous function. When a function is continuous at a point, the limit as the variable approaches that point is simply the value of the function at that point.

• This involves replacing the variable in the expression with the specific number it is approaching.
• In our exercise, if the function is continuous, we plug the value directly into the expression to find the limit.

However, this technique only applies when a function is continuous at the given point. If the function isn't continuous, other limit evaluation methods are required.

Continuous Functions

Continuous functions are ones that have no disruptions in their graphs. For every point within the domain, the function should not have breaks, jumps, or asymptotes.
The expression from our exercise, \( f(x) = \frac{5+\sqrt{x}}{\sqrt{5+x}} \), is composed of basic functions like square roots and polynomials, which are known to be continuous on their domains.
To determine if a function is continuous at a specific point, we must check that the domain includes the point and that there is no denominator of zero or undefined values in the expression.
Once confirmed, continuous functions allow us to use direct substitution to find limits easily.

Expression Simplification

Expression simplification might seem daunting, but it is an essential step for finding clear answers, especially when dealing with limits involving fractions or radicals.
After substituting the point into the expression, simplify it to its most basic form:

• Combine like terms.
• Perform operations such as addition, subtraction, multiplication, or division.

In our solution, substitution led us to \( \frac{5+2}{\sqrt{9}} \) which simplifies to \( \frac{7}{3} \). Simplifying expressions like these helps confirm that our calculations are correct and our limit evaluation makes sense.