Problem 21 - Precalculus

Question

Evaluate the following limits using direct substitution, if possible. If not possible, state why. $$\lim _{x \rightarrow 2} \frac{x^{2}}{5 x-2}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Direct Substitution Check

First, evaluate the expression by substituting the value of the limit directly into the function. Replace every instance of \( x \) with 2 in the function. The function is \( \frac{x^2}{5x - 2} \).

Step 2: Substitute \( x = 2 \) into the Numerator and Denominator

Substitute \( x = 2 \) into the numerator: \( 2^2 = 4 \). Substitute \( x = 2 \) into the denominator: \( 5 \times 2 - 2 = 10 - 2 = 8 \).

Step 3: Compute the Limit Value

Now, substitute these results into the fraction: \( \frac{4}{8} \). Simplify \( \frac{4}{8} \) to \( \frac{1}{2} \).

Step 4: Conclusion

Since substituting \( x = 2 \) into the function does not result in an undefined form (like division by zero), the limit exists and is \( \frac{1}{2} \).

Key Concepts

Direct Substitution

Direct substitution is a relatively straightforward method for finding limits in calculus. When we talk about evaluating the limit of a function as the variable approaches a particular value, the first step is often to substitute this value directly into the function. This process is intuitive: you simply replace the variable with the number you are approaching.

• Begin by identifying the value that the variable is approaching, such as 2 in our case.
• Substitute this value into all instances of the variable in the function.

In some cases, this approach directly yields a numerical answer. An important point to note is that direct substitution might not work if it results in an undefined expression, like division by zero. If your substitution leads to such a case, it indicates a need for further techniques to evaluate the limit.

Numerator and Denominator

In the process of evaluating limits, it's crucial to separately evaluate the numerator and denominator. Each part of the function plays a distinct role in determining the value of the limit. Taking these two steps separately ensures you accurately assess what happens as the variable approaches the given number.

• First, substitute the variable value into the numerator to simplify this part of the function.
• Next, do the same for the denominator.

Consider our example: the function's numerator after substitution becomes 4, because replacing 2 for x in the expression \( x^2 \) results in \( 2^2 = 4 \). The denominator becomes 8, calculated by substituting 2 into \( 5x - 2 \), which results in \( 10 - 2 = 8 \). By processing these parts separately, you can analyze each segment's behavior before combining them.

Simplification

Simplification is the process of reducing expressions to their most basic form. After evaluating both the numerator and denominator, you'll often need to simplify the fraction to clearly understand the limit's value.

• Take the overall fraction formed after substitution and reduce it to the simplest form.
• This involves finding the greatest common divisor (GCD) of both numbers and dividing them.

In our exercise, substituting gave us the fraction \( \frac{4}{8} \). Simplifying this mean you would divide both the numerator and the denominator by their GCD, which is 4, resulting in \( \frac{1}{2} \). Simplification clarifies the result and ensures that you've correctly understood the behavior of the function around the point of interest.

Limit Evaluation

Limit evaluation is the culmination of the process where you determine the behavior of the function as the variable approaches a specific value. This is where you determine if a simpler expression indicates the function's behavior.

• Once the function has been simplified, check the outcome to see if it's a valid numerical result and if the limit exists.
• If direct substitution doesn't yield an undefined form, as in this exercise, conclude with the finalized value of the limit.

For the problem at hand, direct substitution did not produce an undefined form like \( \frac{0}{0} \). Instead, it provided a clear outcome of \( \frac{1}{2} \). The limit evaluation tells us that as \( x \) approaches 2, the function \( \frac{x^2}{5x - 2} \) settles around \( \frac{1}{2} \). By capturing this understanding, you successfully evaluate limits using substitution, complementing algebraic manipulation skills.