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In the world of calculus, the limit of a function is a fundamental concept which helps us understand how a function behaves as its input approaches a particular value. One of the simplest methods for finding limits is through **direct substitution**. This method involves replacing the variable in the function with the value it approaches and evaluating the limit directly.

When you apply direct substitution, you simply plug the value into the function. If the function is well-behaved (i.e., it doesn't have any form of indeterminate expressions like \(\frac{0}{0}\) or \(\infty - \infty\)), this method works smoothly. For polynomial functions, like the one in our exercise, direct substitution often provides a quick and effective way to find limits.

In our example, when \(x = 2\) is substituted into \((x^2 + 1)(2x^2 - 4)\), the expression simplifies without any complications, allowing us to find the limit easily.

Limit evaluation is a crucial technique in calculus used to determine the behavior of a function as the input approaches a certain point. When dealing with limits, it's important to assess whether a method like direct substitution can be employed.

Sometimes, especially with more complicated expressions, limits can present challenges such as indeterminate forms. In such cases, other techniques like factoring, rationalizing, or using L'Hôpital's rule might be necessary.

However, in our exercise, given that both expressions within the limit are polynomials, the evaluation of the limit becomes straightforward. We calculated:

• First, substitute \(x = 2\)
• Next, solve \((2^2 + 1)(2(2)^2 - 4)\)
• Finally, simplify to get \(20\)

Each step confirms smooth evaluation without any hitches, affirming the suitability of direct substitution.

A polynomial function is a mathematical expression consisting of variables and coefficients, structured in terms like \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer. Polynomials are important building blocks in calculus, as they are continuous and smooth, making them amenable to various calculus operations, including taking limits.

In the given exercise, the expression we have is \((x^2 + 1)(2x^2 - 4)\), a polynomial function of degree 4, because the highest power of \(x\) is \(x^4\).

Because polynomial functions are continuous everywhere, evaluating their limits using direct substitution is generally reliable. You simply substitute the value toward which \(x\) is approaching, and the polynomial behaves nicely, without jumps or breaks.

Understanding the continuity of a function is essential when discussing limits. A continuous function has no gaps, jumps, or sudden changes in value. For limits, continuity implies that you can estimate the value of a function as \(x\) approaches a certain point simply by evaluating the function at that point.

Polynomials, such as the one in this exercise, are continuous across their entire domain. This inherent continuity ensures that direct substitution can effectively evaluate limits. It simplifies the process, making it possible to find the limit just by plugging the point into the function without worrying about the complexities that come with discontinuities.

In this problem, the continuity of the polynomial function \((x^2 + 1)(2x^2 - 4)\) guarantees that as \(x\) approaches 2, the function's value converges seamlessly to the calculated limit of 20. Hence, continuity often plays a reassuring role in problems involving limits.