Evaluate each limit and justify your answer. 
lim x→2 (3 / 2x^5−4x^2−50)^4

Verified step by step guidance

1. Identify the type of limit problem: This is a limit of a function as x approaches a specific value.
2. Substitute x = 2 into the function to check if it results in an indeterminate form or a defined value.
3. Calculate the expression inside the parentheses: \(2x^5 - 4x^2 - 50\) by substituting x = 2.
4. Evaluate the expression \(3 / (2x^5 - 4x^2 - 50)\) using the result from the previous step.
5. Raise the result to the power of 4 to find the limit.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating expressions that may not be directly computable at those points.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to undefined expressions or different limit values.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers and are combined using addition, subtraction, and multiplication. In the given limit, the polynomial in the denominator can affect the limit's value, especially if it approaches zero, leading to potential indeterminate forms.