Question

find the following limit or state that it does not exist.

\\\lim_{h \to 0} \frac{(21+h)^2 - 441}{h}\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. \\\lim_{h \to 0} \frac{(21+h)^2 - 441}{h} = \quad\\ (type an exact answer.)
b. the limit does not exist.

Explanation:

Expand the numerator

$$ (21+h)^2 - 441 = 441 + 42h + h^2 - 441 = 42h + h^2 $$

Simplify the rational expression

$$ \frac{42h + h^2}{h} = \frac{h(42 + h)}{h} = 42 + h \quad (h eq 0) $$

Evaluate the limit

$$ \lim_{h \to 0} (42 + h) = 42 $$

Answer:

• A. \(\lim_{h\to0}\frac{(21+h)^2-441}{h} = 42\) (Correct answer)
• B. The limit does not exist.