91Ó°ÊÓ

In calculus, the concept of continuity is fundamental. A function is considered continuous at a specific point if there is no interruption, gap, or jump in its value at that point. To determine if a function is continuous, we check these conditions:

• The function is defined at the point.
• The limit of the function exists as it approaches that point.
• The limit value and the function value at that point are equal.

In the limit problem, we confirm continuity at the point of interest, which is whenever we can simply substitute the approaching value into our function. In the given exercise, the function \( \frac{x}{3} + 2 \) is continuous at \( x = 6 \). There's no division by zero or any other complications, so we can easily calculate the limit directly.

Limits are used to evaluate the behavior of functions as they approach certain inputs. This allows mathematicians to find the value that a function approaches as its variable gets closer to a particular point. In the provided exercise, we are tasked with evaluating \( \lim _{x \rightarrow 6}(x / 3+2) \). This process helps us understand how the function "acts" close to \( x = 6 \), even though the exact numeric value at that point might not be needed.

To evaluate limits effectively, one usually tries to:

• Substitute the value directly if the function is continuous at that point.
• Use algebraic manipulation to simplify the expression if direct substitution leads to an indeterminate form like \( \frac{0}{0} \).
• Apply special limit laws and theorems when necessary.

In simpler cases like this exercise, direct substitution is sufficient, leading directly to the expected answer.

The substitution method is a straightforward technique used to find limits when a function is continuous at the point of interest. If substituting the value the limit approaches into the function doesn't lead to any issues, such as division by zero, then you can directly substitute the number. This is exactly what was done in our original exercise.

We have the expression \( \frac{x}{3} + 2 \). By substituting \( x = 6 \), we transform the expression into \( \frac{6}{3} + 2 \).
Calculating this gives \( 2 + 2 = 4 \).
This confirms that the limit is indeed \( 4 \).

The substitution method is easy and efficient in the right circumstances, and it saved time by allowing us to go directly from the function form to the answer. However, always ensure that the function is continuous at the point you're checking before applying this method.