We are asked to find the slope of the tangent line to the curve $f(x) = 2x^2 - x + 2$ at $x = 2$, and then use that result to write the equation of the tangent line. To do this, we'll use the limit definition of the derivative:

$m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

where $a = 2$. Let's break this process into steps.

Step 1: Compute $f(a + h)$

First, substitute $x = a + h$ into $f(x)$. Here, $a = 2$, so we find $f(2 + h)$:

$f(2 + h) = 2(2 + h)^2 - (2 + h) + 2$

Simplifying the right-hand side:

$= 2(4 + 4h + h^2) - (2 + h) + 2$ $= 2(4 + 4h + h^2) - 2 - h + 2$ $= 8 + 8h + 2h^2 - 2 - h + 2$ $= 8h + 2h^2 + 8 - 2 - h + 2$ $= 2h^2 + 7h + 8$

So, $f(2 + h) = 2h^2 + 7h + 8$.

Step 2: Compute $f(a)$

Next, compute $f(2)$:

$f(2) = 2(2)^2 - 2 + 2 = 8 - 2 + 2 = 8$

Step 3: Apply the limit definition of the derivative

Now, substitute $f(2 + h)$ and $f(2)$ into the limit definition:

$m = \lim_{h \to 0} \frac{f(2 + h) - f(2)}{h}$ $= \lim_{h \to 0} \frac{(2h^2 + 7h + 8) - 8}{h}$ $= \lim_{h \to 0} \frac{2h^2 + 7h}{h}$ $= \lim_{h \to 0} (2h + 7)$

As $h \to 0$, the expression becomes:

$m = 7$

So, the slope of the tangent line at $x = 2$ is $7$.

Step 4: Write the equation of the tangent line

To find the equation of the tangent line, we use the point-slope form of a line:

$y - y_1 = m(x - x_1)$

Here, $m = 7$, $x_1 = 2$, and $y_1 = f(2) = 8$. Substituting these values:

$y - 8 = 7(x - 2)$

Simplifying:

$y - 8 = 7x - 14$ $y = 7x - 6$

Thus, the equation of the tangent line at $x = 2$ is:

$y = 7x - 6$

Would you like a detailed breakdown of any part, or any clarifications?

Here are 5 related questions for further practice:

1. How would you find the slope of the tangent line for $f(x) = 3x^2 - 4x + 1$ at $x = 1$?
2. What does the derivative of a function tell us geometrically?
3. Can you find the derivative of $f(x) = x^3 - 2x + 5$ using the limit definition?
4. How would the tangent line at $x = 0$ differ from the one at $x = 2$ for the same function?
5. How do the limits in calculus relate to the concept of continuity?

Tip: When finding the equation of a tangent line, always double-check both the slope and the point of tangency, as these are critical in the point-slope form of the line.