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

Answer

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

### Step 1: Substitute the value of \( x \)
First, we will substitute \( x = 2 \) directly into the function to see if we can evaluate the limit directly.

\[ \text{Numerator: } x^2 + 5x + 6 = 2^2 + 5(2) + 6 \]

Calculating this gives:

\[ = 4 + 10 + 6 = 20 \]

Now for the denominator:

\[ \text{Denominator: } 2x^2 - 3x = 2(2^2) - 3(2) \]

Calculating this gives:

\[ = 2(4) - 6 = 8 - 6 = 2 \]

### Step 2: Write the limit expression
Now we can write the limit expression with the values we calculated:

\[ \lim_{x \to 2} \frac{x^2 + 5x + 6}{2x^2 - 3x} = \frac{20}{2} \]

### Step 3: Simplify the expression
Now we simplify \( \frac{20}{2} \):

\[ = 10 \]

### Conclusion
Thus, the limit is:

\[ \lim_{x \to 2} \frac{x^2 + 5x + 6}{2x^2 - 3x} = 10 \]

### Final Answer
The final answer is:

\[ \boxed{10} \]