(d) \\( \\lim _ { y \\rightarrow - 2 } \\frac { 2 } { y ^ { 2 } + 2 y } + \\frac { 1 } { y + 2 } \\)
(d) \\( \\lim _ { y \\rightarrow - 2 } \\frac { 2 } { y ^ { 2 } + 2 y } + \\frac { 1 } { y + 2 } \\)
Answer
Explanation:
Step1: Simplify the denominator
Factor (y^{2}+2y=y(y + 2)). Then the original expression (\frac{2}{y^{2}+2y}+\frac{1}{y + 2}=\frac{2}{y(y + 2)}+\frac{1}{y + 2}).
Step2: Find a common denominator
The common denominator is (y(y + 2)). Rewrite (\frac{1}{y + 2}) as (\frac{y}{y(y + 2)}). So the expression becomes (\frac{2+y}{y(y + 2)}).
Step3: Simplify the fraction
Cancel out the common factor ((y + 2)) (since (y\neq - 2) when taking the limit, we can do this simplification). The simplified expression is (\frac{1}{y}).
Step4: Evaluate the limit
Now, find (\lim_{y\rightarrow - 2}\frac{1}{y}). Substitute (y=-2) into (\frac{1}{y}), we get (\frac{1}{-2}=-\frac{1}{2}).
Answer:
(-\frac{1}{2})