Find lim as Δ x to 0 (Δ y)/(Δ x) for y = kx + n

Find $\displaystyle\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$ for $y = kx + n$.

Step 1: Compute the incremented function value

Let $x$ be an arbitrary point and let $\Delta x \neq 0$ be an increment of the argument. Then

\[ y + \Delta y = k(x + \Delta x) + n = kx + k\,\Delta x + n. \]

Step 2: Find the increment $\Delta y$

Subtract $y = kx + n$ from the expression above: $\Delta y = \bigl(kx + k\,\Delta x + n\bigr) - \bigl(kx + n\bigr) = k\,\Delta x$.

Step 3: Form the difference quotient

\[ \frac{\Delta y}{\Delta x} = \frac{k\,\Delta x}{\Delta x} = k, \qquad \Delta x \neq 0. \]

Step 4: Take the limit

The difference quotient equals the constant $k$ for every $\Delta x \neq 0$, so the limit is immediate: $\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} k = k$.

Step 5: Verification

The function $y = kx + n$ is linear, and by the standard rules of differentiation

\[\frac{dy}{dx} = \frac{d}{dx}(kx) + \frac{d}{dx}(n) = k + 0 = k \;\;\checkmark\]

This agrees with the limit computed from first principles.