# Find the following limits or state that they do not exist. lim_{wrightarrow1}frac{acdot1}{(w^2-2)}-frac{1}{(w-1)b}

Find the following limits or state that they do not exist. $\underset{w\to 1}{lim}\frac{a\cdot 1}{\left({w}^{2}-2\right)}-\frac{1}{\left(w-1\right)b}$
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Neelam Wainwright
Given,
$\underset{w\to 1}{lim}\frac{a\cdot 1}{{w}^{2}-2}-\frac{1}{\left(w-1\right)b}$
Rewrite the function:
$\underset{s\to {1}^{-1}}{lim}\left(\frac{a}{{s}^{2}-s}-\frac{b}{s-1}\right)=\underset{s\to {1}^{-1}}{lim}\frac{a-bs}{s\left(s-1\right)}$
The limit of product/quotient is the product/quotient of limits:
$\underset{s\to {1}^{-1}}{lim}\frac{a-bs}{s\left(s-1\right)}=\underset{s->{1}^{-1}}{lim}\frac{a-bs}{s\cdot \underset{s->{1}^{-1}}{lim}\frac{1}{s-1}}$
The limit of a quotient is the quotient of limits:
$\underset{s\to {1}^{-1}}{lim}\frac{\left(a-bs\right)}{s\cdot \underset{s\to {1}^{-1}}{lim}\frac{1}{s-1}}=\underset{s\to {1}^{-1}}{lim}\frac{1}{s-1}\cdot \underset{s\to {1}^{-1}}{lim}\frac{a-bs}{\underset{s\to {1}^{-1}}{lim}s}$
Substitete the variable with the value:
$\underset{s\to {1}^{-1}}{lim}\frac{1}{s-1}\cdot \underset{s\to {1}^{-1}}{lim}\frac{\left(a-bs\right)}{\underset{s->{1}^{-1}}{lim}s}=\underset{s\to {1}^{-1}}{lim}\frac{\frac{1}{\left(s-1\right)\cdot \left(a-b\right)}}{\underset{s\to {1}^{-1}}{lim}s}$
Substitete the variable with the value:
$\left(a-b\right)\underset{s\to {1}^{-1}}{lim}\frac{1}{s-1}\cdot \underset{s\to {1}^{-1}}{lim}{s}^{-1}=\left(a-b\right)\underset{s\to {1}^{-1}}{lim}\frac{1}{s-1}\cdot {1}^{-1}$
The function decreases without a bound:
$\underset{s\to {1}^{-1}}{lim}\frac{1}{s-1}=-\mathrm{\infty }$
Therefore,
$\underset{s\to {1}^{-1}}{lim}\frac{a}{{s}^{2}-s}-\frac{b}{s-1}=-\mathrm{\infty }\left(a-b\right)$
Rewrite the function:
$\underset{s\to {1}^{1}}{lim}\frac{a}{{s}^{2}-s}-\frac{b}{s-1}=\underset{s\to {1}^{1}}{lim}\frac{a-bs}{s\left(s-1\right)}$
The limit of a product/quotient is the product/quotient of limits:
$\underset{s\to {1}^{1}}{lim}\frac{a-bs}{s\left(s-1\right)}=\underset{s\to {1}^{1}}{lim}\frac{a-bs}{s\cdot \underset{s\to {1}^{1}}{lim}}\left(\frac{1}{s-1}\right)$
The limit of quotient is the quotient of limits:
$\underset{s->{1}^{1}}{lim}\frac{1}{s-1}\cdot \underset{s\to {1}^{1}}{lim}\frac{a-bs}{s}=\underset{s\to {1}^{1}}{lim}\frac{1}{s-1}\underset{s\to {1}^{1}}{lim}\frac{a-bs}{\underset{s\to {1}^{1}}{lim}s}$
Substitute variable with the value: