Question

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 4 } ( x + 3 ) ^ { 1998 }$$

Short answer

The limit of the function as \(x\) approaches \(-4\) is \(1\).

Step-by-step solution

Understanding Substitution

In mathematics, substitution method means replacing a variable by its actual value. To evaluate a limit through substitution, simply plug the target value of x, which in this case is \(-4\), directly into the equation.

Perform Substitution

Take the equation: \( ( x + 3 ) ^ { 1998 }\). Substitute \(-4\) in place of \(x\) in the function, therefore it ends up being \( ( -4 + 3 ) ^ { 1998 }\). This simplifies to \( -1 ^ { 1998 }\).

Calculate the Result

Next step is calculating \( -1 ^ { 1998 }\). We know that any negative number to an even power is positive. Thus \( -1 ^ { 1998 }\) simplifies to \(1\).

Graphical Representation

On graphing \( ( x + 3 ) ^ { 1998 }\), it's observed that as \(x\) approaches \(-4\), the value of function approaches \(1\) which supports the limit calculated above.

Key concepts

Evaluating Limits

The process of evaluating limits is fundamental in calculus. It involves finding the value that a function approaches as the input (or 'x') gets close to some number. For example, when we have \( \lim _{x \rightarrow -4} (x + 3)^{1998} \) and we want to find out what value the expression approaches as 'x' gets close to -4.

First, we look to see if the function is continuous at 'x' equals -4. If it's continuous, we can usually just substitute the value of 'x' directly into the function. This is what's often referred to as the 'direct substitution' method. When direct substitution results in an expression like 0/0, further techniques such as factoring, rationalizing, or applying special limits may be necessary.

Substitution Method in Calculus

The substitution method in calculus is a straightforward way to evaluate limits and functions. This method works by simply plugging in the value to which 'x' is approaching into the function. If the result is a real number, that number is the limit.

For instance, in \( (x + 3)^{1998} \) we substitute -4 for 'x' to get \( (-4 + 3)^{1998} \) which simplifies to \( -1^{1998} \) because -4 plus 3 is -1. This simple substitution process is a powerful tool and often the first method attempted when evaluating limits. However, it's critical to note that this method applies best when the function is not undefined at the point of interest.

Negative Exponents

When dealing with negative exponents, remember that they indicate the inverse of the base raised to the corresponding positive exponent. However, in the context of evaluating limits, what often matters more is the parity of the exponent - whether it is even or odd. An even exponent will result in a positive result when the base is negative, as in the solution to our exercise \( -1^{1998} \) which equals to 1, since 1998 is an even number.

Understanding this concept is crucial in simplifying expressions before applying the substitution method, especially when we encounter negative bases raised to high powers.

Graphical Representation of Limits

The graphical representation of limits can be very helpful in understanding the behavior of functions as 'x' approaches a specific value. By plotting the function on a graph, we make it easier to visualize how the function behaves near the point of interest. When we graph the function \( (x + 3)^{1998} \) and observe as 'x' approaches -4, we see that the curve near this point gets closer to 1.

Graphs can also reveal whether the limits from the left and right are equal, which means the overall limit exists at that point. They serve as an excellent visual aid that confirms our algebraic calculations when evaluating limits, and can also help us spot nuances such as asymptotic behavior or oscillation that pure algebraic methods might not make immediately apparent.