details\nno additional details were added for this assignment.\nhw5 the limit laws (target l4; §2.3)\nscore…
details\nno additional details were added for this assignment.\nhw5 the limit laws (target l4; §2.3)\nscore: 4/13 answered: 4/13\nquestion 5\nevaluate the limit: $lim_{h\rightarrow0}\frac{(9 + h)^{2}-81}{h}$\nquestion help: video message instructor
Answer
Explanation:
Step1: Expand the numerator
Expand ((9 + h)^2-81). Using the formula ((a + b)^2=a^{2}+2ab + b^{2}), where (a = 9) and (b=h), we get ((9 + h)^2-81=(81 + 18h+h^{2})-81=18h + h^{2}).
Step2: Simplify the fraction
The original limit (\lim_{h\rightarrow0}\frac{(9 + h)^2-81}{h}) becomes (\lim_{h\rightarrow0}\frac{18h+h^{2}}{h}). Factor out an (h) from the numerator: (\lim_{h\rightarrow0}\frac{h(18 + h)}{h}). Cancel out the common - factor (h) (since (h\neq0) when taking the limit), and we have (\lim_{h\rightarrow0}(18 + h)).
Step3: Evaluate the limit
Substitute (h = 0) into (18 + h). We get (18+0 = 18).
Answer:
18