Question

(d) \\( \lim _ { y \
ightarrow - 2 } \frac { 2 } { y ^ { 2 } + 2 y } + \frac { 1 } { y + 2 } \\)

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 \(y
eq - 2\) when taking the limit, we can do this simplification). The simplified expression is \(\frac{1}{y}\).

Step4: Evaluate the limit

Now, find \(\lim_{y
ightarrow - 2}\frac{1}{y}\). Substitute \(y=-2\) into \(\frac{1}{y}\), we get \(\frac{1}{-2}=-\frac{1}{2}\).

Answer:

\(-\frac{1}{2}\)