Evaluate `lim_(x to 2) (x^2+x+2)/(x^3+1)`

Answer

To evaluate the limit \( \lim_{x \to 2} \frac{x^2 + x + 2}{x^3 + 1} \), we will follow these steps:

### Step 1: Substitute the value of \( x \)
We start by substituting \( x = 2 \) directly into the expression.

\[ \frac{2^2 + 2 + 2}{2^3 + 1} \]

### Step 2: Calculate the numerator
Now, we calculate the numerator:

\[ 2^2 = 4 \]
\[ 4 + 2 + 2 = 8 \]

So, the numerator becomes \( 8 \).

### Step 3: Calculate the denominator
Next, we calculate the denominator:

\[ 2^3 = 8 \]
\[ 8 + 1 = 9 \]

So, the denominator becomes \( 9 \).

### Step 4: Form the limit expression
Now we can form the limit expression:

\[ \frac{8}{9} \]

### Step 5: Conclusion
Thus, the limit is:

\[ \lim_{x \to 2} \frac{x^2 + x + 2}{x^3 + 1} = \frac{8}{9} \]

### Final Answer
The final answer is \( \frac{8}{9} \).

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