Problem 22 - APEX Calculus

Question

Evaluate the given limit. $$ \lim _{x \rightarrow 1} \frac{2 x-2}{x+4} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Substitute the limit value into the expression

To evaluate the limit, start by substituting the value that x is approaching, which is 1, directly into the expression. Substitute x = 1 into \( \frac{2x-2}{x+4} \), resulting in \( \frac{2(1)-2}{1+4} = \frac{0}{5} \).

Step 2: Simplify the expression

Since direct substitution resulted in a determinate form, simplify the expression if possible. In this case, \( \frac{0}{5} \) is already simplified as 0.

Step 3: Conclude the limit evaluation

Since the resulting expression is \( \frac{0}{5} \), the limit can be concluded as 0.

Key Concepts

Direct Substitution

When evaluating the limit of a function as it approaches a certain value, one of the first methods to consider is direct substitution. This technique involves directly inserting the point into the limit function to see if it results in a simplifiable expression.

• Start by identifying the point that the variable is approaching, which is often given directly in the exercise. In our case, it is when \(x\) approaches 1.
• Substitute this value into the function. Here, you substitute \(x = 1\) into \( \frac{2x-2}{x+4} \).
  This calculation gives \( \frac{2(1)-2}{1+4} = \frac{0}{5} \).

Direct substitution is quick when applicable, especially when it yields a determinate form.

Determinate Form

The term "determinate form" refers to the outcome of a substitution in a limit evaluation, where the expression results in a specific, concrete value. This is opposite of an indeterminate form, which is one of the unreadable expressions like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

• If, after substitution, the expression resolves to a clear, finite value, it is a determinate form.
  In this example: \(\frac{0}{5}\) is a determinable end solution, because the "0" implies a result to be exact and does not require further calculation.
• Determinate forms mean that additional techniques like factoring, rationalizing, or otherwise manipulating the function might not be necessary, as there are no uncertainties left in the final result.

Understanding whether the result is determinate or indeterminate is crucial in confirming whether the solution is complete.

Simplifying Expressions

Simplification is a key principle in solving mathematical problems, including those involving limits. Once you've substituted and found a result, it’s always good practice to check if it can be simplified further.

• First, ensure that the numerator and denominator cannot be reduced further.
  For the expression \(\frac{0}{5}\), the numerator is already zero, which makes the entire expression zero.
• Simplification ensures that the expression is in its most basic form, revealing insights into the function's behavior.
  In many cases, like this example, thoughtful simplification confirms the limit as \(0\).

Ultimately, by exercising the practice of simplification, mathematical expressions become more approachable and understandable.