Question

Can someone help me with some algebra?

Answer 1

★ Oblique Asymptote ★

$f(x) = \frac{x {}^{2} + 5x + 6}{x - 4} \\ f(x) = (x - 4)(x + 9) + 42$
Hence , oblique Asymptote is obtained simultaneously by the Quotient of the function obtained ,
HENCE , oblique Asymptote is
$y = x + 9$

Answer 2

Answer:

y=x+9.

Step-by-step explanation:

the function is $\frac{x^{2}+5x+6}{x-4}$. Now, the equation of the oblique asymptote is a function y= mx + b where

m = $\lim_{x \to \infty} \frac{f(x)}{x}$

and b = $\lim_{x \to \infty} f(x)-mx$

So,

m =  $\lim_{x \to \infty} \frac{\frac{x^{2}+5x+6}{x-4}}{x}$

= $\lim_{x \to \infty} \frac{x^{2}+5x+6}{x^{2}-4x}= 1.$

and,

b = $\lim_{x \to \infty}\frac{x^{2}+5x+6}{x-4}-x$

=  $\lim_{x \to \infty}\frac{x^{2}+5x+6-x^{2}+4x}{x-4}$

=  $\lim_{x \to \infty}\frac{9x+6}{x-4}= 9.$

Then, the equation of the oblique asymptote is y= x+9.