Question

Which statement comparing the oblique asymptotes of the functions f(x) and g(x) is true?

The oblique asymptote for g(x) is steeper than for f(x).
Both functions have the same oblique asymptote.
The oblique asymptote for f(x) is steeper than for g(x).
Both functions have parallel oblique asymptotes.

Answer 1

I can’t tell which statement is true without the definitions or graphs of f(x) and g(x). Please post the formulas or the graph.

How to decide once you give them:
- If f and g are rational with deg(numerator) = deg(denominator) + 1, do polynomial long division to get the oblique asymptote y = mx + b for each. The slope m is the ratio of leading coefficients.
- Equivalently, for an oblique asymptote y = mx + b,
- m = lim_{x→∞} f(x)/x,
- b = lim_{x→∞} (f(x) − m x).
- Compare the slopes m_f and m_g:
- If m_f = m_g and b_f = b_g → “Both functions have the same oblique asymptote.”
- If m_f = m_g but b_f ≠ b_g → “Both functions have parallel oblique asymptotes.”
- If |m_f| > |m_g⟩ → “The oblique asymptote for f(x) is steeper than for g(x).” (steeper means larger absolute slope)
- If |m_g| > |m_f| → “The oblique asymptote for g(x) is steeper than for f(x).”

Send the functions and I’ll compute and pick the correct statement.