Question

Evaluate each limit and justify your answer. $$\lim _{x \rightarrow 3} \sqrt{x^{2}+7}$$

Short answer

Answer: The value of the limit is 4.

Step-by-step solution

Identify the limit and the function

The limit is given as x approaches 3, and the function is $$\sqrt{x^2 + 7}$$. So we want to evaluate $$\lim_{x \rightarrow 3} \sqrt{x^{2}+7}$$.

Substitute the limit value into the function

Substitute x by 3 in the function and simplify the expression: $$\sqrt{(3)^{2} + 7}$$

Calculate the expression

Calculate the square and simplify the expression: $$\sqrt{9 + 7}$$ $$\sqrt{16}$$

Find the square root

Find the square root of 16: $$\sqrt{16} = 4$$ So, the limit of the function $$\sqrt{x^{2}+7}$$ as $$x \rightarrow 3$$ is 4. Therefore, $$\lim_{x \rightarrow 3} \sqrt{x^{2}+7} = 4$$.

Key concepts

Square Root Function

The square root function is a vital concept in mathematics. It refers to the operation of finding a number which, when multiplied by itself, yields the given number. For example, the square root of 16 is 4. This is because 4 times 4 equals 16. The square root symbol, often a radical sign, appears as \( \sqrt{} \). Within the realm of functions, it can transform variables in expressions. Understanding this function requires recognizing that it only pertains to non-negative numbers. This means that for a function \( f(x) = \sqrt{x} \), \( x \) must be greater than or equal to zero. Any x-value that makes the expression under the square root negative is not within the function's domain. In our exercise, the square root function \( \sqrt{x^2 + 7} \) is used, where the expression inside the radical must remain non-negative for all values of \( x \). Knowing the properties of square root functions makes evaluating limits involving them more intuitive.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that examines the behavior of a function as the input values approach a specific value. The goal is to determine the value that a function approaches as the inputs get arbitrarily close to a given number. To evaluate limits, we often assess the function by substitution, simplification, or considering the function's behavior near the point of interest. The limit can provide insights into the continuity and behavior of functions at specific points or even infinity. In the problem \( \lim_{x \rightarrow 3} \sqrt{x^2 + 7} \), we evaluate the limit by observing what happens to \( \sqrt{x^2 + 7} \) as \( x \) approaches 3. Performing the calculation step-by-step helps ensure we understand each part of the transformation process. The evaluation confirms that the function approaches a specific numerical value, in this case 4, as \( x \) approaches the value of 3.

Direct Substitution

Direct substitution is a straightforward method used in limit evaluation, allowing us to plug the value of \( x \) directly into the function. When the function is continuous at the point being examined, direct substitution is often the simplest and most effective approach to finding the limit. Consider the function \( \sqrt{x^2 + 7} \) and the corresponding limit \( \lim_{x \rightarrow 3} \sqrt{x^2 + 7} \). By merely substituting \( x = 3 \) directly into the expression, we can evaluate the limit without complicated manipulations.

• First, replace \( x \) with 3, which leads to \( \sqrt{3^2 + 7} \).
• Then simplify it to \( \sqrt{9 + 7} \), which is just \( \sqrt{16} \).
• Finally, calculate the square root to arrive at the result 4.

This process reveals the limit in an uncomplicated manner. Direct substitution works smoothly here because the function \( \sqrt{x^2 + 7} \) is continuous at the point \( x = 3 \).