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To find the limit of the given rational function:

\[
\lim_{{x \to 3}} \frac{2x^2 + 4x - 5}{x^2 - 5x + 6}
\]

### Step 1: Evaluate the direct substitution
We first check what happens if we directly substitute \(x = 3\) into the function:

- For the numerator: 
  \[
  2(3)^2 + 4(3) - 5 = 2(9) + 12 - 5 = 18 + 12 - 5 = 25
  \]
- For the denominator:
  \[
  (3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0
  \]

Since the denominator is zero, we have an indeterminate form \(\frac{25}{0}\), so we can't evaluate directly. We need to simplify the expression.

### Step 2: Factor both numerator and denominator

- The numerator \(2x^2 + 4x - 5\) can be factored as:
  \[
  2x^2 + 4x - 5 = (2x - 1)(x + 5)
  \]

- The denominator \(x^2 - 5x + 6\) can be factored as:
  \[
  x^2 - 5x + 6 = (x - 2)(x - 3)
  \]

Thus, the expression becomes:
\[
\frac{(2x - 1)(x + 5)}{(x - 2)(x - 3)}
\]

### Step 3: Cancel common factors
There are no common factors to cancel, but now we can substitute \(x = 3\) directly into the simplified expression:

\[
\frac{(2(3) - 1)(3 + 5)}{(3 - 2)(3 - 3)} = \frac{(6 - 1)(8)}{(1)(0)} = \frac{5 \cdot 8}{1 \cdot 0}
\]

This still results in division by zero, implying the limit does not exist. The function approaches infinity or negative infinity as \(x\) approaches 3.

### Conclusion:
The limit does not exist because the denominator tends to zero while the numerator tends to a non-zero number, indicating that the function has a vertical asymptote at \(x = 3\).

Would you like more details or further explanation?

Here are five related questions you might find helpful:
1. What is the definition of a vertical asymptote?
2. How can we determine the behavior of the function as it approaches the asymptote?
3. How do you handle limits involving indeterminate forms like \(\frac{0}{0}\)?
4. Can we apply L'Hôpital's Rule to this problem, and how would it change the solution?
5. What happens if the numerator also tends to zero when finding limits?

**Tip:** Always check for factorization when encountering indeterminate forms like \(\frac{0}{0}\) in limit problems.

Find the limit lim(x→3) (2x^2 + 4x - 5)/(x^2 - 5x + 6)

Understand how to find the limit of the rational function lim(x→3) (2x^2 + 4x - 5)/(x^2 - 5x + 6). This example demonstrates key calculus concepts like limits, factorization, and handling indeterminate forms.

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