Evaluate `lim_(x to 0) [(3x^(2) + 4x + 5)/(x^(2) - 2x + 3)]`

Answer

To evaluate the limit \(\lim_{x \to 0} \frac{3x^2 + 4x + 5}{x^2 - 2x + 3}\), we will follow the steps outlined in the video.

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

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

\[ f(0) = \frac{3(0)^2 + 4(0) + 5}{(0)^2 - 2(0) + 3} \]

Calculating the numerator:

\[ 3(0)^2 + 4(0) + 5 = 0 + 0 + 5 = 5 \]

Calculating the denominator:

\[ (0)^2 - 2(0) + 3 = 0 - 0 + 3 = 3 \]

So we have:

\[ f(0) = \frac{5}{3} \]

### Step 2: Check if the result is finite

Since \(\frac{5}{3}\) is a finite value, we do not have an indeterminate form (like \(0/0\) or \(\infty/\infty\)). Therefore, we can conclude that:

\[ \lim_{x \to 0} \frac{3x^2 + 4x + 5}{x^2 - 2x + 3} = \frac{5}{3} \]

### Final Answer

Thus, the limit is:

\[ \frac{5}{3} \]

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