Evaluating the Limit of 2 - x1/(1-x) as x Approaches 1
In calculus, evaluating limits of indeterminate forms, such as 1^∞, is a common yet challenging task. This article explores a particular example that involves the limit of the expression 2 - x1/(1-x) as x approaches 1. We will demonstrate how to evaluate this limit using various methods, including direct substitution, logarithmic differentiation, and L'H?pital's Rule.
Introduction to the Problem
The expression we are dealing with is:
L limx→1 2 - x1/(1-x)
This expression is an example of an indeterminate form 1^∞, which requires careful manipulation to yield a meaningful result. We will explore several methods to evaluate this limit.
First Method: Direct Exponential Transformation
Step 1: The expression can be transformed using the property of exponents and logarithms:
L limx→1 e(1/(1-x)2 - x - 1)
Step 2: Evaluating the exponent as x approaches 1:
e1 e
Second Method: Using the 1^∞ Form
Step 1: Recognizing the form of the expression:
L limx→1 2 - x1/(1-x)
Step 2: Converting the expression using the exponential function:
exp[limx→1(1/(1-x)(2 - x - 1))]
Step 3: Simplifying the exponent:
exp[limx→1(1 - x)/(1 - x)] exp[1]
Step 4: Final result:
e
Third Method: Long Method with Logarithmic Differentiation
Step 1: Let L limx→1 2 - x(1/(1-x)).
Step 2: Taking the natural logarithm on both sides:
ln L limx→1 (ln(2 - x))/(1 - x)
Step 3: Applying L'H?pital's Rule:
limx→1 (1)/(2 - x) 1
Step 4: Therefore:
ln L 1
Step 5: Exponentiating both sides:
L e
Final Method: Indeterminate Form Simplification
Since the expression f(x)g(x) results in the form of 1^∞ at x→1, the limit can be evaluated as:
L elimx→1(1/(1-x)(2 - x - 1))
elimx→1(1 - x)/(1 - x)
e1
e
Conclusion
Through these various methods, we have demonstrated that the limit of the expression 2 - x1/(1-x) as x approaches 1 is equal to the mathematical constant e. The methods used include direct exponential transformation, recognizing indeterminate forms, logarithmic differentiation, and L'H?pital's Rule. Understanding these techniques is crucial for handling similar limit problems in calculus.