Evaluate :
`lim_(xto 0 ) ((x+2)/(x+1))^(x+3)`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \left(\frac{x+2}{x+1}\right)^{x+3}, \]

we can follow these steps:

### Step 1: Substitute \( x = 0 \)

We start by substituting \( x = 0 \) into the expression:

\[ \frac{x+2}{x+1} \text{ becomes } \frac{0+2}{0+1} = \frac{2}{1} = 2. \]

### Step 2: Substitute \( x = 0 \) in the exponent

Next, we substitute \( x = 0 \) into the exponent \( x + 3 \):

\[ x + 3 \text{ becomes } 0 + 3 = 3. \]

### Step 3: Combine the results

Now we can combine the results from Step 1 and Step 2:

\[ \left(\frac{x+2}{x+1}\right)^{x+3} \text{ becomes } 2^3. \]

### Step 4: Calculate \( 2^3 \)

Finally, we calculate \( 2^3 \):

\[ 2^3 = 8. \]

### Final Answer

Thus, the limit is

\[ \lim_{x \to 0} \left(\frac{x+2}{x+1}\right)^{x+3} = 8. \]

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