Problem 57 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to -2}\ \dfrac{5x+3}{2x-9}$$

Step-by-Step Solution

Answer

After carrying out the above steps, we find that the limit of the function as \( x \) approaches -2 is -1.

Step 1: Substitution

The value of \( x \) given is \( -2 \). So, substitute \( x = -2 \) into the given expression \( \dfrac{5x+3}{2x-9} \)

Step 2: Compute the Result

After substituting \( -2 \) for \( x \) , we will compute the numerator and the denominator separately (i.e calculate \( 5(-2)+3 \) and \( 2(-2)-9 \) respectively ), then divide to find the answer.

Key Concepts

Direct Substitution

Direct substitution is one of the simplest ways to compute limits. The method involves directly replacing the variable in the expression with the number it is approaching. In our exercise, we are dealing with the limit \( \lim_{x \to -2}\ \frac{5x+3}{2x-9} \). Here, \( x \) approaches \(-2\).
To use direct substitution, we substitute \( x = -2 \) directly into the expression. This method relies on the function being nice enough for this substitution nature, meaning it doesn't result in a division by zero or other undefined situations.
When successful, direct substitution efficiently provides the limit without further manipulation. It works particularly well when the function in question is continuous at the point of interest.

Numerator and Denominator

Understanding how the numerator and denominator separately affect the result is crucial when working with fractional expressions. In our expression \( \frac{5x+3}{2x-9} \), the numerator is \( 5x + 3 \) and the denominator is \( 2x - 9 \). Each portion has its effect on the outcome, and their interaction is essential.
When computing the limit, substitute \( x = -2 \) into both the numerator and denominator:

• Numerator: \( 5(-2) + 3 = -10 + 3 = -7 \)
• Denominator: \( 2(-2) - 9 = -4 - 9 = -13 \)

Working separately on both parts helps to avoid confusion and ensure clarity in calculations. Once these values are understood, they can be combined to evaluate the whole expression's limit.

Computing Limits

Once all substitutions and individual computations of the numerator and denominator are done, it's time to find the overall limit value. Calculating the final limit involves simply dividing the result of the numerator by the result of the denominator.

• Numerator value: \(-7\)
• Denominator value: \(-13\)

Therefore, the limit is computed as \( \frac{-7}{-13} \). Simplifying this fraction gives \( \frac{7}{13} \), as negative signs cancel each other out.
Through this step-by-step breakdown of direct substitution and individual calculations, you obtain your solution: \( \frac{7}{13} \). This process reaffirms how mathematical operations require precision and careful handling of expressions to achieve the correct result.