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Sketch the Function $y=\frac{x}{x-1}$

(1) Domain R / $\{1\}$

(2) Intercepts: $y$ -intercept $(x=0) \rightarrow y=\frac{0}{1}=0$

$x$ -intercopt $(y=0) \rightarrow 0=\frac{x}{x-1} \rightarrow x=0$

$(0,0)$ Intercept

(3) Assymptotes: $H \cdot A$:

$\lim _{x \rightarrow \infty} \frac{x}{x-1}=1$

$\lim _{x \rightarrow -\infty} \frac{x}{x-1}=1$

Assymptotes: $V \cdot A$:

$\lim _{x \rightarrow 1^{+}} \frac{x}{x-1}=+\infty$ 1.1

$\lim _{x \rightarrow 1^{-}} \frac{x}{x-1}=-\infty$ 0.9

(4) $y^{\prime}=\frac{(x-1)-x}{(x-1)^{2}}=\frac{-1}{(x-1)^{2}}$

$y^{\prime}(x)=0 \Rightarrow \frac{-1}{(-1)^{2}}=0 \quad$ Can't be

$y^{\prime}(x) \quad D N E \quad x=1$

$y^{\prime}(2)=-1<0$

$-y$ is Decreasing on $(-\infty, 1) \cup(1, \infty)$

- No lacal Max or Min

- Concavity and Inflection points:

$y^{\prime}(x)= \frac{-1}{(x-1)^{2}}$

$y^{\prime \prime}=\frac{-2(x-1)(-1)}{((x-1)^{2})^{2}}=\frac{2(x-1)}{(x-1)^{4}}=\frac{2}{(x-1)^{3}}$

$y^{\prime \prime}=0 \Rightarrow \frac{2}{(x-1)^{3}}=0$ can't be

$y^{\prime \prime}(x) D N E \Rightarrow x=1$

$y^{\prime \prime}(-1)=-$

$y^{\prime \prime}(2)=+$