Evaluate the following limits :
`lim_(x to 2)(x^(3)-2x^(2))/(x^(2)-5x+6)`

Answer

To evaluate the limit \( \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} \), we will follow these steps:

### Step 1: Factor the numerator and denominator
First, we will factor both the numerator and the denominator.

The numerator is:
\[ x^3 - 2x^2 = x^2(x - 2) \]

The denominator is:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Rewrite the limit
Now we can rewrite the limit using the factored forms:
\[ \lim_{x \to 2} \frac{x^2(x - 2)}{(x - 2)(x - 3)} \]

### Step 3: Cancel common factors
We can cancel the common factor \( (x - 2) \) from the numerator and denominator:
\[ \lim_{x \to 2} \frac{x^2}{x - 3} \]

### Step 4: Substitute the limit value
Now we can substitute \( x = 2 \) into the simplified expression:
\[ \frac{2^2}{2 - 3} = \frac{4}{-1} = -4 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 2} \frac{x^3 - 2x^2}{x^2 - 5x + 6} = -4 \]

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