Section 15.3

Oblique asymptotes

viz

y=mx+b\text{ is an oblique asymptote} \Leftrightarrow \lim_{x\to+\infty}[f(x)-(mx+b)]=0\ \text{or}\ \lim_{x\to-\infty}[f(x)-(mx+b)]=0

The oblique asymptote may exist only at $+\infty$ or at $-\infty$ .

\text{For }x\to+\infty:\ m_+=\lim_{x\to+\infty}\frac{f(x)}{x},\ b_+=\lim_{x\to+\infty}[f(x)-m_+x];\quad \text{for }x\to-\infty:\ m_-=\lim_{x\to-\infty}\frac{f(x)}{x},\ b_-=\lim_{x\to-\infty}[f(x)-m_-x]

Slope and intercept for oblique asymptotes (apply separately at $+\infty$ and $-\infty$ ).

$\text{for }x\to-\infty:\ m_-=\lim_{x\to-\infty}\frac{f(x)}{x},\ b_-=\lim_{x\to-\infty}[f(x)-m_-x]$

Loading section…

Section 15.3

viz

y=mx+b\text{ is an oblique asymptote} \Leftrightarrow \lim_{x\to+\infty}[f(x)-(mx+b)]=0\ \text{or}\ \lim_{x\to-\infty}[f(x)-(mx+b)]=0

The oblique asymptote may exist only at $+\infty$ or at $-\infty$ .

\text{For }x\to+\infty:\ m_+=\lim_{x\to+\infty}\frac{f(x)}{x},\ b_+=\lim_{x\to+\infty}[f(x)-m_+x];\quad \text{for }x\to-\infty:\ m_-=\lim_{x\to-\infty}\frac{f(x)}{x},\ b_-=\lim_{x\to-\infty}[f(x)-m_-x]

Slope and intercept for oblique asymptotes (apply separately at $+\infty$ and $-\infty$ ).

$\text{for }x\to-\infty:\ m_-=\lim_{x\to-\infty}\frac{f(x)}{x},\ b_-=\lim_{x\to-\infty}[f(x)-m_-x]$