Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer.
f(x)= √x−2; a=1

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To determine if a function is continuous at a point \(a\), we need to check three conditions: (1) \(f(a)\) is defined, (2) \(\lim_{{x \to a}} f(x)\) exists, and (3) \(\lim_{{x \to a}} f(x) = f(a)\).

Substitute \(a = 1\) into the function \(f(x) = \sqrt{x} - 2\). Calculate \(f(1)\) to see if it is defined.

Find the limit of \(f(x) = \sqrt{x} - 2\) as \(x\) approaches 1. This involves substituting \(x = 1\) into the expression under the limit.

Check if the value of \(f(1)\) is equal to the limit \(\lim_{{x \to 1}} f(x)\).

Based on the results from the previous steps, determine if all three conditions for continuity are satisfied at \(a = 1\). If they are, the function is continuous at \(a = 1\); otherwise, it is not.

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Continuity of Functions

A function is continuous at a point 'a' if three conditions are met: the function is defined at 'a', the limit of the function as 'x' approaches 'a' exists, and the limit equals the function's value at 'a'. This concept is fundamental in calculus as it ensures that there are no breaks, jumps, or holes in the graph of the function at that point.

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. For continuity, it is essential to evaluate the limit of the function as 'x' approaches 'a' and confirm that it matches the function's value at 'a'. This concept helps in understanding how functions behave near specific points.

Square Root Function

The square root function, denoted as √x, is defined only for non-negative values of 'x'. This means that for the function f(x) = √x - 2 to be continuous at 'a', 'a' must be within the domain of the square root function. Understanding the domain is crucial for determining continuity, especially when evaluating functions involving roots.

Determine whether the following functions are continuous at a. Use the continuity checklist to justify your answer. f(x)= √x−2; a=1