Finding the Limit as y Approaches 1 of (1/y - 1/1)/(y - 1)

In the field of mathematics, the concept of limits is a fundamental topic that plays a crucial role in understanding and analyzing various functions and their behaviors. One of the common exercises in high school mathematics is to find the limit of a given expression as a variable approaches a specific value.

In this article, we will explore the process of finding the limit as y approaches 1 of the expression (1/y - 1/1)/(y - 1).

Understanding the Expression

The given expression (1/y - 1/1)/(y - 1) can be broken down into two parts:

1. 1/y - 1/1: This part represents the difference between the reciprocal of the variable y and the reciprocal of the constant 1.
2. y - 1: This part represents the difference between the variable y and the constant 1.

The overall expression is the quotient of these two parts, which is (1/y - 1/1)/(y - 1).

Finding the Limit

To find the limit as y approaches 1, we need to evaluate the expression as y gets closer and closer to 1. This can be done by following these steps:

1. Substitute y = 1 into the expression:

• (1/1 - 1/1)/(1 - 1) = (0/0)

  This results in an indeterminate form, which means we cannot directly evaluate the expression by substituting y = 1.

1. Simplify the expression:

• (1/y - 1/1)/(y - 1) = (1/y - 1)/(y - 1)

1. Apply algebraic manipulation:

• Multiply the numerator and denominator by the conjugate of the denominator: (1/y - 1)/(y - 1) = (1/y - 1) * (1/1) / ((y - 1) * 1/1)
• Simplify the expression: (1/y - 1) * (1/1) / ((y - 1) * 1/1) = (1/y - 1) / (y - 1)

1. Evaluate the limit as y approaches 1:

• Substitute y = 1 into the simplified expression: (1/1 - 1) / (1 - 1) = 0/0

  This is still an indeterminate form, so we need to use the concept of limits to find the actual limit value.

1. Use the limit definition:

• The limit of the expression (1/y - 1) / (y - 1) as y approaches 1 can be found by applying the limit definition:
• lim(y->1) (1/y - 1) / (y - 1) = (1/1 - 1) / (1 - 1) = 0/0

1. Apply L'Hôpital's rule:

• L'Hôpital's rule states that if the limit of a fraction results in an indeterminate form (0/0 or ∞/∞), we can differentiate the numerator and denominator separately and then evaluate the limit of the resulting fraction.
• Differentiating the numerator: d/dy (1/y - 1) = -1/y^2
• Differentiating the denominator: d/dy (y - 1) = 1
• Applying L'Hôpital's rule: lim(y->1) (1/y - 1) / (y - 1) = lim(y->1) (-1/y^2) / 1 = -1/1^2 = -1

Therefore, the limit as y approaches 1 of the expression (1/y - 1/1)/(y - 1) is -1.

In conclusion, by applying the concepts of limits, algebraic manipulation, and L'Hôpital's rule, we have successfully found the limit as y approaches 1 of the given expression (1/y - 1/1)/(y - 1).