Solving Limits Involving Exponentials and Indeterminate Forms

Understanding and solving limits that involve exponentials can be a challenging task. This article aims to simplify the process by breaking down a specific example and explaining the steps to solve it effectively. We will cover the problem: lim_{x→1} [4^x - 3^{1/2x-2}].

Introduction to the Problem

The given limit is lim_{x→1} [4^x - 3^{1/2x-2}]. Initially, this limit appears to be in an indeterminate form, as evaluating it at (x 1) directly leads to 1^∞, which is an indeterminate form. Therefore, we need to apply techniques such as the L'Hospital's Rule to find the limit.

Solving the Limit Step-by-Step

Step 1: Convert the Expression to an Exponential Form To simplify the expression, we first introduce a natural logarithm to transform the limit. Consider the natural logarithm of the expression:

lnL lim_{x→1} ln [4^x - 3^{1/2x-2}]

Using the properties of logarithms, we can rewrite this as:

lnL lim_{x→1} ln [4^x - 3] / (2x - 2)

This form, lnL lim_{x→1} [ln 4^x - 3 / 2x - 2], is an indeterminate form 0/0, which allows us to apply L'Hospital's Rule.

Step 2: Apply L'Hospital's Rule Applying L'Hospital's Rule, we differentiate the numerator and the denominator with respect to (x):

lnL lim_{x→1} [d/dx ln 4^x - 3 / d/dx 2x - 2]

Further simplifying, we get:

lnL lim_{x→1} [4^x log 4 / (4^x - 3) / 2]

Since (4^1 4) and (3^1/2 * 1 - 2 3/2 - 2 -1/2), we can evaluate the limit at (x 1):

lnL log 4 / 2 log 2^2 log 16

Therefore, we have:

L e^{log 16} 16

Alternative Approach

Another approach to solving the same limit is to use the concept of indeterminate forms and exponential functions directly. Let’s revisit the limit in exponential form:

L lim_{x→1} [4^x - 3^{1/2x-2}]

We can convert this to an exponential form:

L e^{lim_{x→1} [ln (4^x - 3^{1/2x-2})]}

Using logarithm properties, we get:

L e^{lim_{x→1} [1/2x - 2 ln 4^x - 3]}

Recognizing the form as L'Hospital's Rule, we differentiate the numerator and denominator:

L e^{lim_{x→1} [(4^x log 4) / (2 (4^x - 3))]}

Evaluating the limit at (x 1):

L e^{log 16} 16

Conclusion

The final result is 16. This step-by-step approach illustrates how to manage and simplify limits involving indeterminate forms, particularly those involving exponentials. Understanding these techniques is crucial for tackling similar problems in calculus and mathematical analysis.

Related Keywords

The key terms associated with this topic include: limit, exponential function, indeterminate form.

This article provides a comprehensive guide for students and professionals dealing with similar mathematical problems.