Questions: lim as x approaches 1 of (x^3 - 7x^2 + 12x) / (4 - x)

Transcript text: $\lim _{x \rightarrow 1} \frac{x^{3}-7 x^{2}+12 x}{4-x}$

Solution

Solution Steps

Step 1: Evaluate the Limit Directly

We start by substituting $ x = 1 $ into the expression $ \frac{x^{3} - 7x^{2} + 12x}{4 - x} $: \[ \frac{1^{3} - 7 \cdot 1^{2} + 12 \cdot 1}{4 - 1} = \frac{1 - 7 + 12}{3} = \frac{6}{3} = 2 \]

Step 2: Check for Indeterminate Form

Next, we check if substituting $ x = 1 $ results in an indeterminate form. The numerator becomes $ 1 - 7 + 12 = 6 $ and the denominator becomes $ 4 - 1 = 3 $, which is not an indeterminate form. Thus, we can conclude that the limit exists.

Step 3: State the Limit

Since the direct substitution does not yield an indeterminate form, we find that: \[ \lim_{x \rightarrow 1} \frac{x^{3} - 7x^{2} + 12x}{4 - x} = 2 \]