Problem 52 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to -2}\ (x^3 -6x+5)$$

Step-by-Step Solution

Answer

The limit of the function \( x^3 -6x+5 \) as \( x \) approaches -2 is 9.

Step 1: Substitute \( x \) with -2

We begin by substituting \( x \) with -2 in the function \( x^3 -6x+5 \). This gives us \( (-2)^3-6(-2)+5 \).

Step 2: Calculate \( (-2)^3 \)

Next, calculate \( (-2)^3 \), which equals -8. So, our expression now looks like \( -8-6(-2)+5 \).

Step 3: Calculate \( -6(-2) \)

Now, calculate \( -6(-2) \), which equals 12. So, our result thus far is \( -8+12+5 \).

Step 4: Simplify the Expression

Finally, we simplify the result. \( -8+12+5 \) equals 9.

Key Concepts

Direct Substitution Method

When dealing with calculus and the concept of limits, the direct substitution method is your first line of attack. It's both the simplest and often the quickest way to find the limit of a function as you approach a specific value for a variable.

Here's how it plays out: start by taking your function and replace the variable with the value that you're approaching. For polynomial functions, this is often all you have to do. Why? Because polynomials are continuous functions, meaning, they don't have breaks, jumps, or holes in their graphs at real numbers. So, if you're trying to find the limit as the variable approaches any real number, the function's value at that number is the limit.

It becomes trickier when the function isn't defined at that point, or the operation leads to an indeterminate form like 0/0. For those cases, you'd need more advanced techniques like factoring, conjugate multiplication, or L'Hôpital's Rule. However, our focus here is when direct substitution works flawlessly as with polynomials.

Limit of a Function

The limit of a function is a fundamental concept in calculus. It describes what value the function approaches as the input (usually denoted by x) approaches some value. Think of a limit as the function's 'goal' as x gets closer and closer to a particular point, even if it never quite reaches it.

Understanding limits allows mathematicians to deal with situations where a function doesn't have a clear value at a point, often due to discontinuities or points of undefined behavior. For instance, what happens to the value of a function as it approaches an asymptote or a hole in the graph? That's where limits come in.

In the exercise given, there's no such confusion — the limit exists and is simply the value of the function at x = -2. But the underlying idea of limits still provides the justification for why direct substitution is the correct strategy to employ in this scenario.

Polynomial Functions

Polynomial functions are algebraic expressions that involve only non-negative integer powers of x. They are composed of terms which are made up of a coefficient and a variable. An example of a polynomial function is the one seen in the exercise, which is a cubic polynomial because the highest power of x is three.

One of the amazing properties of polynomial functions is their continuity everywhere on the real number line. In other words, their graphs are unbroken lines or smooth curves. This property guarantees that for polynomial functions, you can find the limit at any point simply by plugging in the value of x, which relates back to the direct substitution method.

Remember that despite their simplicity, polynomial functions can model a wide range of real-world phenomena, from simple linear trends to complex behaviors described by higher-degree polynomials. This versatility, combined with their mathematical 'niceness,' makes them a cornerstone of calculus.