Intercept of Asymptote

Note that we can write the given equation as

$y = \frac{ x^{2 }- 2x + 7x + 6 }{x - 2 }$ $y = \frac { x ( x - 2) + 7x -14 + 20 } { x - 2}$ $y = x + \frac {7(x - 2 ) + 20 } { x - 2}$ $y = x + 7 + \frac{ 20 } { x - 2 }$

Observe that when the absolute value of $x$ is very very large, like for example $x= 1000000$ or $x = - 3000000$ , the fraction term on the right hand side becomes very small and y is approximately equal to $x + 7$ . When $x \to\pm \infty$ , $y \to x + 7$ . Thus, $y = x + 7$ is an oblique asymptote of the given curve.

The y intercept of $y = x + 7$ is the value of $y$ when $x = 0$ , that is $y = \boxed{ 7 }$ $_\square$

Put $y=mx+c$ in the given equation and since the equation we need is not vertical therefore we can multiply both sides by $(x-2)$ and get $(mx+c)(x-2)=x^2+5x+6$

$\implies (m-1)x^2-(2m-c+5)x-(2c+6)=0$

This equation has double root at $x -> \infty$

So put coeff of $x^2$ = coefficient of $x$ = $0$

$m-1=2m-c+5=0$

$\implies m-1=0,2m-c+5=0$

$m=1,c=7$

So the y-intercept is $c=\boxed{7}$

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