Questions: Is the following limit form determinate or indeterminate?

Transcript text: Is the following limit form determinate or indeterminate?

Solution

Solution Steps

To determine if a limit is in a determinate or indeterminate form, we need to evaluate the expression as it approaches a certain value. Common indeterminate forms include 0/0, ∞/∞, 0*∞, ∞-∞, 0^0, ∞^0, and 1^∞. If the expression simplifies to one of these forms, it is indeterminate; otherwise, it is determinate.

Step 1: Define the Expression and the Limit

We are given the expression \(\frac{x^2 - 1}{x - 1}\) and need to evaluate the limit as \(x\) approaches 1.

Step 2: Simplify the Expression

The expression \(\frac{x^2 - 1}{x - 1}\) can be simplified by factoring the numerator: \[ x^2 - 1 = (x - 1)(x + 1) \] Thus, the expression becomes: \[ \frac{(x - 1)(x + 1)}{x - 1} \] For \(x \neq 1\), this simplifies to: \[ x + 1 \]

Step 3: Evaluate the Limit

Now, we evaluate the limit of the simplified expression \(x + 1\) as \(x\) approaches 1: \[ \lim_{x \to 1} (x + 1) = 1 + 1 = 2 \]

Step 4: Determine the Form

Since the limit evaluates to a finite number, it is a determinate form.