Problem 47 - Calculus for Biology and Medicine

Question

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 3} \frac{3 x^{2}+1}{2 x-3} $$

Step-by-Step Solution

Answer

The limit is \( \frac{28}{3} \).

Step 1: Identify the Direct Substitution

The first step in evaluating the limit is to replace the variable in the function with the limit point. In this case, we substitute 3 into the function: \( \lim _{x \rightarrow 3} \frac{3 \times 3^2 + 1}{2 \times 3 - 3} \).

Step 2: Simplify the Numerator and Denominator

Compute the values after substitution in both numerator and denominator. For the numerator: \( 3 \times 3^2 + 1 = 3 \times 9 + 1 = 27 + 1 = 28 \). For the denominator: \( 2 \times 3 - 3 = 6 - 3 = 3 \).

Step 3: Evaluate the Limit Expression

Plug the simplified results back into the fraction: \( \frac{28}{3} \).

Step 4: Conclude the Limit Evaluation

After simplifying the fraction, we see the limit \( \lim _{x \rightarrow 3} \frac{3x^2 + 1}{2x - 3} \) evaluates directly to \( \frac{28}{3} \), since there is no indeterminate form.

Key Concepts

Direct Substitution

Direct substitution is the easiest method for finding limits. It involves replacing the variable in the function with the value that the variable is approaching. In the given limit problem, we are trying to evaluate \( \lim _{x \rightarrow 3} \frac{3x^2 + 1}{2x - 3} \).

• We start by substituting \( x = 3 \) into the expression.
• This means we calculate: \( \frac{3 \cdot 3^2 + 1}{2 \cdot 3 - 3} \).

Direct substitution is the first rule of thumb in limit evaluation because it can quickly reveal whether the limit exists or if further simplification is needed. If substituting results in a defined value (not an indeterminate form like \( \frac{0}{0} \)), the process is straightforward and complete.

Simplifying Expressions

After substituting the variable with the given value, the next step is simplifying the expression. This involves simplifying the numerator and the denominator separately before combining them. In our example:

• The numerator is \( 3x^2 + 1 \), which after plugging in \( x = 3 \), simplifies to \( 3 \times 9 + 1 = 28 \).
• The denominator is \( 2x - 3 \), and with \( x = 3 \), it becomes \( 2 \times 3 - 3 = 3 \).
• Thus, the expression becomes \( \frac{28}{3} \).

Simplifying expressions correctly is crucial as it directly impacts the outcome of the limit evaluation. Incorrect simplification can lead to wrong conclusions about the limit.

Evaluating Limits

Evaluating limits is the final step where we determine if the function approaches a particular value as the variable gets infinitely close to a certain point. Once you have simplified the expressions from substitution:

• The simplified expression \( \frac{28}{3} \) is already in its final form.
• Since no further simplification is required, this fraction is our limit.

In this problem, because the substitution did not result in an indeterminate form (like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \)), we could conclude the evaluation simply with the fraction result. This direct conclusion shows that the function \( \frac{3x^2 + 1}{2x - 3} \) approaches \( \frac{28}{3} \) as \( x \) approaches 3.