Solving Limits Through Substitution and Factoring: A Comprehensive Guide
Understanding and solving complex limits can be challenging, especially when traditional methods are not immediately apparent. In this article, we will walk you through a detailed process of solving limits using substitution and factoring techniques. Let's explore a specific problem and break down the solution step-by-step.
Problem Statement
Consider the following limit problem:
$$displaystyle lim_{{x to 2}} frac{sqrt[4]{frac{2}{x}} - x - 1}{left(sqrt[4]{frac{2}{x}} - xright)^2 - 1}$$
This problem might seem intimidating at first glance, but with a strategic approach, it can be easily resolved.
Solution
Step 1: Substitution
One effective method to simplify the problem is by making a substitution. Let's define:
$$t sqrt[4]{frac{2}{x}}$$
This implies:
$$x 2/t^4$$
Using this substitution, we can rewrite the limit as:
$$displaystyle lim_{{t to 1}} frac{t - frac{2}{t^4} - 1}{t^2 - frac{4}{t^3} frac{4}{t^8} - 1}$$
Simplifying the numerator and the denominator:
Numerator:
$$t - frac{2}{t^4} - 1 frac{t^5 t^4 - 2}{t^4}$$
Denominator:
$$t^2 - frac{4}{t^3} frac{4}{t^8} - 1 frac{t^{10} - t^8 - 4t^5 4}{t^8}$$
Therefore, the limit becomes:
$$displaystyle lim_{{t to 1}} frac{frac{t^5 t^4 - 2}{t^4}}{frac{t^{10} - t^8 - 4t^5 4}{t^8}} lim_{{t to 1}} frac{t^4 (t^5 t^4 - 2)}{t^{10} - t^8 - 4t^5 4}$$
This is in the form of an indeterminate form 0/0, which allows us to factor out (t-1) from both the numerator and the denominator:
Numerator after factoring:
$$t^5 t^4 - 2 (t - 1) (t^4 2t^3 2t^2 2t 2)$$
Denominator after factoring:
$$t^{10} - t^8 - 4t^5 4 (t - 1) (t^9 - t^8 - 4t^4 - 4t^3 - 4t^2 - 4t - 4)$$
Therefore, the limit simplifies to:
$$displaystyle lim_{{t to 1}} frac{(t^4 2t^3 2t^2 2t 2)}{(t^9 - t^8 - 4t^4 - 4t^3 - 4t^2 - 4t - 4)} frac{1 2 2 2 2}{1 - 1 - 4 - 4 - 4 - 4 - 4} -frac{1}{2}$$
Alternative Method
Another method to solve this problem is by directly rewriting the initial expression. Let:
$$L displaystyle lim_{{x to 2}} frac{sqrt[4]{frac{2}{x}} - x - 1}{(sqrt[4]{frac{2}{x}} - x)^2 - 1}$$
This can be rewritten as:
$$L displaystyle lim_{{x to 2}} frac{sqrt[4]{frac{2}{x}} - x - 1}{(sqrt[4]{frac{2}{x}} - x - 1)(sqrt[4]{frac{2}{x}} - x 1)}$$
Cancelling the common factor:
$$L displaystyle lim_{{x to 2}} frac{1}{(sqrt[4]{frac{2}{x}} - x 1)}$$
Evaluating the limit:
$$L -frac{1}{2}$$
Conclusion
In conclusion, solving complex limits requires a systematic approach, often involving substitution and factoring techniques. By breaking down the problem and simplifying step-by-step, we can find the solution. As shown in the examples above, the key is to identify patterns and make strategic substitutions to reveal the underlying simplicity of the problem.