91Ó°ÊÓ

When dealing with limits in calculus, we're typically figuring out what value a function approaches as the variable gets closer to a certain point. Evaluating limits is essential for understanding the behavior of functions, particularly when direct computation at a point would be tricky or impossible. In our given problem, we want to find out what the expression \( \frac{x^2 - 2x}{x + 1} \) approaches as \( x \) gets close to 3. This process is known as limit evaluation.

There are various techniques for evaluating limits, such as:

• Direct Substitution: Plugging in the value of \( x \) directly, if it doesn't lead to indeterminate forms like \( \frac{0}{0} \).
• Factoring: Useful for simplifying expressions to remove troublesome terms.
• Rationalizing: Applying this technique assists when dealing with square roots.
• Using Conjugates: Often applied for expressions involving radicals.

For our problem, direct substitution helps us quickly find the limit since inserting \( x = 3 \) doesn’t result in indeterminate or undefined expressions. This straightforward method saves time and effort, leading us rapidly to the solution.

The substitution method is one of the simplest and most direct ways to evaluate limits. This involves swapping the close-to limit value into our function to see what it turns into.

For our problem, we substituted \( x = 3 \) into the expression \( \frac{x^2 - 2x}{x+1} \). Here's a breakdown of the method:

• Replace all instances of \( x \) with the limit value, so in our case, every \( x \) is replaced with 3.
• Solve the resulting expression to see if it simplifies to a finite value.

This approach works nicely when substitution results in a defined numerical outcome. It’s also a great initial test to quickly ascertain a limit's value before attempting more complex methods.

In our case, by applying the substitution method, we efficiently determined that the expression doesn't become undefined, guiding us directly to the solution.

Simplifying expressions is a crucial skill when working with limits in calculus. Once substitution has taken place, the expression often requires simplification to reach the final answer.

In the given exercise, after substituting 3 for \( x \), we need to simplify \( \frac{3^2 - 2 \times 3}{3 + 1} \). Here’s how we simplified:

• Calculate the numerator: \( 3^2 = 9 \) and \(-2 \times 3 = 6\), which further reduces to \( 9 - 6 = 3 \).
• Compute the denominator: \( 3 + 1 = 4 \).
• Combine these results: simplifying to \( \frac{3}{4} \).

Simplifying aids in clearly interpreting results and ensures we can express our answer in its most precise form. This step is crucial for making sure the computation's outcome is accurate and understandable. Once simplified, the limit expression became simple to interpret, and we identified the solution's value as \( \frac{3}{4} \). This crucial step confirms our calculation's integrity and offers a clean, concise conclusion to finding the limit.