Let's evaluate the given limit:

$\lim_{x \to 10} \frac{x^2 - x + 3}{x - 10}$

Step 1: Direct Substitution

Substituting $x = 10$:

$\frac{10^2 - 10 + 3}{10 - 10} = \frac{100 - 10 + 3}{0} = \frac{93}{0}$

Since division by zero is undefined, the function has a vertical asymptote at $x = 10$, meaning the limit does not exist in a finite sense.

Step 2: Checking One-Sided Limits

Since the denominator $x - 10$ approaches zero as $x \to 10$, let's analyze the behavior from both sides:

• As $x \to 10^+$ (approaching 10 from the right), $x - 10$ is slightly positive.
• As $x \to 10^-$ (approaching 10 from the left), $x - 10$ is slightly negative.

The numerator, $x^2 - x + 3 = x(x - 1) + 3$, is always positive for values around $x = 10$. Since the denominator changes signs on either side, the function approaches $+\infty$ on one side and $-\infty$ on the other.

Conclusion:

Since the left-hand and right-hand limits do not approach the same value, the limit does not exist (DNE).