Evaluate each limit. 


$\underset{x\to 3}{\lim }\sqrt{x^2+7}$

Verified step by step guidance

1. Identify the limit expression: \( \lim_{x \to 3} \sqrt{x^2 + 7} \).
2. Recognize that this is a direct substitution problem since the function \( \sqrt{x^2 + 7} \) is continuous at \( x = 3 \).
3. Substitute \( x = 3 \) directly into the expression: \( \sqrt{3^2 + 7} \).
4. Simplify the expression inside the square root: \( 3^2 + 7 = 9 + 7 \).
5. Calculate the square root of the simplified expression: \( \sqrt{16} \).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 3.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is essential for evaluating limits, as it allows us to directly substitute the value into the function if it is continuous. The function in the limit question, √(x² + 7), is continuous everywhere since it is a polynomial under a square root.

Substitution in Limits

Substitution is a technique used in limit evaluation where you replace the variable in the function with the value it approaches. If the function is continuous at that point, this method yields the limit directly. For the given limit, substituting x = 3 into the function √(x² + 7) will provide the limit value without further manipulation.