If f(x) = x^2 +5x, find lim delta x f (x + delta x ) - f (x) / delta x

Question

anonymous · Answer

$$\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$
You left it out, but I'm assuming $\Delta x$ is approaching zero, since this looks like you're supposed to find the derivative of the given $f(x)$ using the limit definition. Also, for the sake of brevity, I'm going to denote $h=\Delta x$.
$$\begin{align*}\lim_{h\to0}\frac{f(x+h)-f(x)}{h}&=\lim_{h\to0}\frac{\bigg((x+h)^2+5(x+h)\bigg)-\bigg(x^2+5x\bigg)}{h}\
&=\lim_{h\to0}\frac{\bigg(x^2+2xh+h^2+5x+5h\bigg)-\bigg(x^2+5x\bigg)}{h}\
&=\lim_{h\to0}\frac{2xh+h^2+5h}{h}\
&=\lim_{h\to0}(2x+h+5)\
&=\cdots
\end{align*}$$