Problem 19 - Calculus and its applications

Question

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3} $$

Step-by-Step Solution

Answer

The limit is 6.

Step 1: Substitute and Check if Indeterminate

First, substitute \(x = 3\) into the expression \(\frac{x^2 - 9}{x - 3}\). This gives: \(\frac{3^2 - 9}{3 - 3} = \frac{9 - 9}{0} = \frac{0}{0}\), which is an indeterminate form. Hence, we need to simplify the expression.

Step 2: Factor the Numerator

The expression \(x^2 - 9\) can be factored as a difference of squares: \(x^2 - 9 = (x + 3)(x - 3)\). This simplifies the function to: \(\frac{(x + 3)(x - 3)}{x - 3}\).

Step 3: Cancel Out Common Factors

Observe that both the numerator and the denominator have a common factor \((x - 3)\). Cancel this common factor to simplify the expression to \(x + 3\).

Step 4: Substitute Again to Find the Limit

Now, substitute \(x = 3\) into the simplified expression \(x + 3\): \(3 + 3 = 6\). Therefore, the limit is 6.

Key Concepts

Indeterminate Forms

When evaluating the limits in calculus, you'll often encounter expressions that result in forms like \( \frac{0}{0} \). This is called an "indeterminate form". It suggests that further work is needed to determine the limit of the expression. Finding an indeterminate form means the expression doesn't yet tell us much about the limit value.

• Why is it indeterminate? Because \( \frac{0}{0} \) can imply a variety of results depending on how the terms simplify.
• This form indicates that both the numerator and the denominator are approaching zero as \( x \) approaches the limit value.
• It's a signal to explore other methods, like simplification or algebraic manipulation, to find a workable form.

Once you find an indeterminate form like \( \frac{0}{0} \), your goal is to manipulate the expression to remove the indeterminacy and properly evaluate the limit. In our original exercise, after initial substitution, we discovered the form \( \frac{0}{0} \) and moved forward with further steps to simplify the function.

Difference of Squares

A difference of squares is an algebraic expression in the form \( a^2 - b^2 \). It can be factored into \((a - b)(a + b)\), which is a key identity you will frequently use in calculus.

• Useful for simplifying expressions that initially appear complex.
• The key to using this rule effectively is recognizing patterns in the expression, like \( x^2 - 9 \) in our exercise.
• This forms the basis of manipulating indeterminate limits, as it helps to cancel out terms.

In our example, \( x^2 - 9 \) was factored using the difference of squares technique:\[x^2 - 9 = (x - 3)(x + 3)\]This facilitated the cancellation of \( (x - 3) \) from the numerator and denominator, which is crucial for simplifying limits. Remembering this pattern can make it easier to handle similar expressions in calculus.

Simplifying Rational Expressions

Simplifying rational expressions is about making an algebraic fraction less complex by canceling common factors in the numerator and the denominator. This is especially useful when dealing with indeterminate forms and limits.

• First, identify common factors. Look closely at both the numerator and denominator.
• Use algebraic identities, such as the difference of squares, to rewrite polynomials.
• Cancel common factors to simplify the expression, as allowable under the rules of algebra (do not cancel terms that make the denominator zero!).

In our example, we simplified the expression \( \frac{(x + 3)(x - 3)}{x - 3} \) by canceling \( (x - 3) \). This left us with \( x + 3 \), a much simpler expression.
Once simplified, substituting back in the limit provides a clear picture of the function's behavior near the limit point, leading directly to the result of the limit.