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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

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


$$-y$$ 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

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

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

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


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