Understanding the Limit as x Approaches 9 for the Function f(x) (sqrt{x}-3)/(x-9) and Its Value

In calculus, determining limits for complex functions can often involve a careful and systematic approach. Let's delve into how we can find the limit of the function g(x) (sqrt(x)-3)/(x-9) as x approaches 9. The result is not zero, but 1/6. This article will explore the steps involved in this process and provide a deeper understanding of the underlying mathematical principles.

Step-by-Step Solution: Simplifying the Function

Let's start by examining the given function: g(x) (sqrt(x)-3)/(x-9). The direct substitution of x 9 leads to an indeterminate form, 0/0. Therefore, we need to manipulate the function before taking the limit.

First, we rationalize the numerator by multiplying the numerator and the denominator by the conjugate of the numerator, which is (sqrt(x) 3). This step will help us eliminate the square root in the numerator.

The expression becomes:

g(x) (sqrt(x)-3)/(x-9) * (sqrt(x) 3)/(sqrt(x) 3)

Upon simplification, the numerator turns into a difference of squares:

(sqrt(x) - 3)(sqrt(x) 3) x - 9

Thus, the function now appears as:

g(x) (x - 9)/(x - 9)(sqrt(x) 3)

We can now cancel out the common factor (x-9) from the numerator and the denominator:

g(x) 1/(sqrt(x) 3)

Next, we take the limit as x approaches 9:

lim_{xto 9} 1/(sqrt(x) 3) 1/(sqrt(9) 3) 1/(3 3) 1/6

Explanation of the Mathematics

The key to solving this problem lies in the rationalization step. By multiplying the numerator and the denominator by the conjugate of the numerator, we convert the indeterminate form into a determinate one. The function becomes simpler and more manageable when we cancel out the common factors.

The rationalization process is a common technique used in calculus to handle expressions involving square roots. It simplifies the algebraic manipulation and makes it easier to compute the limit.

Why Isn't the Limit Zero?

It might seem intuitive to think that the function (sqrt(x) - 3)/(x - 9) would simplify to 0 as x approaches 9, but that is not the case. The reason lies in the nature of the function. The numerator (sqrt(x) - 3) and the denominator (x - 9) both approach zero as x approaches 9, but the relationship between them is more complex than a simple linear dependence.

When we rationalize the expression, we effectively transform the original function into a new form that is well-defined at x 9. The limit of 1/(sqrt(x) 3) as x approaches 9 is 1/6, which is not zero.

Conclusion

The limit of the function g(x) (sqrt(x) - 3)/(x - 9) as x approaches 9 is not zero, but 1/6. This result is obtained through a series of algebraic manipulations, including rationalization and simplification. Understanding these steps and the underlying mathematical principles will be beneficial for solving similar problems in calculus.

Further Reading and Resources

Articles related to limits in calculus Interactive online tutorials on rationalization and limit solving Practice problems involving limits with square roots

Keywords: limit, calculus, function, approach, x-axis