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hw5 the limit laws (target l4; §2.3)
score: 10/13 answered: 10/13
question 11
evaluate the limit
lim_{y→3} \frac{3(y^{2}-1)}{7y^{2}(y - 1)^{3}}
question help: message instructor

Explanation:

Step1: Expand the expressions

First, expand the numerator $3(y^{2}-1)=3y^{2}-3$ and the denominator $7y^{2}(y - 1)^{3}=7y^{2}(y^{3}-3y^{2}+3y - 1)=7y^{5}-21y^{4}+21y^{3}-7y^{2}$.

Step2: Substitute $y = 3$ into the limit expression

We have $\lim_{y
ightarrow3}\frac{3(y^{2}-1)}{7y^{2}(y - 1)^{3}}=\frac{3(3^{2}-1)}{7\times3^{2}(3 - 1)^{3}}$.

Step3: Calculate the numerator

$3(3^{2}-1)=3\times(9 - 1)=3\times8 = 24$.

Step4: Calculate the denominator

$7\times3^{2}(3 - 1)^{3}=7\times9\times8=504$.

Step5: Simplify the fraction

$\frac{24}{504}=\frac{1}{21}$.

Answer:

$\frac{1}{21}$