Question

Evaluate the limit, if it exists. lim (h - > 0) ((-7 + h)^2 - 49) / h

Answer 1

Expand everything in the limit:

$\displaystyle\lim_{h\to0}\frac{(-7+h)^2-49}h=\lim_{h\to0}\frac{(49-14h+h^2)-49}h=\lim_{h\to0}\frac{h^2-14h}h$

We have $h$ approaching 0, and in particular $h\neq0$, so we can cancel a factor in the numerator and denominator:

$\displaystyle\lim_{h\to0}\frac{h^2-14h}h=\lim_{h\to0}(h-14)=\boxed{-14}$

Alternatively, if you already know about derivatives, consider the function $f(x)=x^2$, whose derivative is $f'(x)=2x$.

Using the limit definition, we have

$f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h=\lim_{h\to0}\frac{(x+h)^2-x^2}h$

which is exactly the original limit with $x=-7$. The derivative is $2x$, so the value of the limit is, again, -14.