\lim_{x\rightarrow{\inf}}{\frac{x^2-4}{2+x-4x^2}} mean?The notation \lim_{x\rightarrow{\inf}}{\frac{x^2-4}{2+x-4x^2}} represents the limit of a function as x approaches infinity. In other words, it represents the value that the function approaches as x gets larger and larger.
To solve this type of limit problem, you can use algebraic manipulation, factoring, and other mathematical techniques to simplify the expression and then use the rules of limits to evaluate the limit. In this specific problem, you can factor out an x^2 term from both the numerator and denominator to simplify the expression.
The final answer to this limit problem is 0. By simplifying the expression and evaluating the limit using the rules of limits, we can see that the limit approaches 0 as x approaches infinity.
Limit problems are important because they allow us to understand the behavior of a function as the input (x) approaches a certain value. This information can be useful in a variety of scientific and mathematical applications, such as determining the maximum or minimum value of a function or analyzing the convergence of a series.
Some common mistakes to avoid when solving limit problems include forgetting to factor out common terms, incorrect application of the rules of limits, and not considering the behavior of the function as x approaches the limit value from both sides. It is important to carefully follow the steps of solving a limit problem and double-check your work to avoid these mistakes.