Problem 56 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to 4}\ \dfrac{x-1}{x^2 +2x+3}$$

Step-by-Step Solution

Answer

The limit of \(\dfrac{x-1}{x^2 +2x+3}\) as \(x\) approaches 4 is \(\dfrac{1}{9}\)

Step 1: Identify the limit

In this exercise, we are asked to find the limit of \(\dfrac{x-1}{x^2 +2x+3}\) as \(x\) approaches 4.

Step 2: Direct Substitution

The method of direct substitution involves substituting the number \(x\) is approaching into the function. In this case, we are to substitute 4 into \(x\), meaning \(\dfrac{4-1}{4^2 +2*4 +3}\).

Step 3: Calculate the Result

Now, calculate the quotient above to get the limit. After the calculation, the result is \(\dfrac{3}{27}\).

Step 4: Simplify the Result

To simplify the resulting fraction, we divide both the numerator and denominator by their greatest common divisor, which is 3. Hence, the result becomes \(\dfrac{1}{9}\).

Key Concepts

Direct Substitution Method

When attempting to find the limit of a function as a certain variable approaches a particular value, one of the most straightforward techniques available is the direct substitution method. This method is particularly useful when the function is well-behaved, meaning it’s continuous and does not produce an undefined or indeterminate form when the variable is substituted.

Let's consider an example. Suppose you have a function such as \(f(x) = \dfrac{x-1}{x^2 +2x+3}\), and you need to find the limit as \(x\) approaches 4. You can apply direct substitution by plugging 4 straight into the function: \(f(4) = \dfrac{4-1}{4^2 +2*4+3}\). If this direct computation yields a definite number, then you've found the limit. Otherwise, you may need to use alternative methods if the substitution leads to an indeterminate form like 0/0 or ∞/∞.

Limit of a Function

The concept of a limit is foundational in calculus and precalculus. It describes the behavior of a function as the input approaches a certain value. In technical terms, the limit of a function at a particular point refers to the value that the function approaches as the input gets arbitrarily close to that point. It’s important to understand that the limit refers to the value the function approaches, not necessarily the value the function actually reaches.

In our example \( \lim_{x \to 4} \dfrac{x-1}{x^2 +2x+3}\), it’s about determining what value the fraction approaches as \(x\) gets closer and closer to 4. By using the direct substitution method, we determined that the value approached is 1/9.

Simplifying Fractions

Simplification is an essential skill in mathematics, particularly when dealing with fractions. Simplifying a fraction means reducing it to its lowest possible terms without changing its value. This typically involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

Let's illustrate this with our limit fraction \(3/27\). To simplify it, find the GCD of 3 and 27, which is 3. Then, divide both the top and bottom by 3, yielding \(1/9\). Even though \(3/27\) and \(1/9\) may look different, they represent the same value. When simplifying, your goal is to make the fraction as straightforward as possible. This not only makes your results tidier but can also make further calculations much easier.

Precalculus

Precalculus is a course of study that bridges the concepts of algebra and calculus. It covers a variety of topics like functions, complex numbers, polynomials, and, importantly, the concept of limits. A good grasp on precalculus prepares students for the more abstract and demanding challenges found in calculus.

In the context of our limit problem, precalculus provides the tools and understanding necessary to deal with functions and their limits. Direct substitution, which we have been discussing, is a concept often introduced in precalculus as a stepping stone to more advanced calculus concepts. By mastering techniques such as simplifying fractions and direct substitution, students are building a solid foundation for tackling the more conceptual and intricate aspects of calculus.