91Ó°ÊÓ

Understanding the concept of limits is essential in calculus as it allows us to study the behavior of functions as they approach a certain value. For example, the limit problem \( \lim _{x \rightarrow 1} \frac{\sqrt{x+3}}{\sqrt[3]{x+7}} \) asks us to find out what value the function \( \frac{\sqrt{x+3}}{\sqrt[3]{x+7}} \) gets closer to, as \( x \) approaches 1.

A common method to evaluate limits, as shown in the textbook solution, is to directly substitute the approached value into the function, but this only works if the function is continuous at that point. In this case, after substitution, we get a rational number which confirms that the function is indeed continuous at \( x = 1 \) and directly substituting is valid. There are instances, however, where direct substitution isn't possible, like when it results in an undefined form such as \( 0/0 \) or \( \infty/\infty \). In those cases, we resort to alternate techniques like factoring, rationalization, or applying L'Hôpital's rule.

Continuity of a function at a point means that there is no interruption in the graph of the function at that point—it's a smooth curve with no holes, jumps, or breaks. A function \( f(x) \) is continuous at a point \( x=a \) if these three conditions are met:

• The function \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( f(a) \).

In solving the provided exercise, we verified the function's continuity by showing that, at \( x=1 \) the function does not create any undetermined form or discontinuity. As we found that the limit as \( x \) approaches 1 is exactly the value of the function at that point, we confirmed the function's continuity.

Radical simplification involves rewriting expressions containing roots in a simpler or more convenient form. For instance, square roots and cubic roots of perfect squares or cubes can be simplified to whole numbers. Looking at our exercise, the expression within the limit had a square root and a cubic root: \( \sqrt{4} \) simplifies to 2 and \( \sqrt[3]{8} \) simplifies to 2, as both 4 and 8 are perfect powers (2 squared and 2 cubed respectively).

The process of simplifying allows us to see more clearly the behavior of the function as the variable approaches a particular value, and in many cases, it aids in determining the limit without more complex calculations. Simplified radicals are foundational to limits since they often allow for the direct substitution method used in the provided step-by-step solution.