Asymptote Finder
Algebra, Functions
Intro: Given a function (usually rational), identifies vertical, horizontal, and oblique asymptotes using limits and degree comparisons.
Worked example
- Find the vertical and oblique asymptotes of $f(x) = \dfrac{2x^2 + 3}{x − 1}$.
- We have $f(x)=\dfrac{2x^2+3}{x-1}$.
- Vertical asymptotes occur where the denominator is zero and the numerator is nonzero.
- Set denominator to zero: $x-1=0 \Rightarrow x=1$. Since numerator at x=1 is $2(1)^2+3=5 \neq 0$, there is a vertical asymptote at $x=1$.
- Next, compare degrees of numerator and denominator. Numerator has degree 2; denominator has degree 1.
- Since degree of numerator is exactly one more than degree of denominator, we expect an oblique (slant) asymptote.
- Perform polynomial long division: divide $2x^2+0x+3$ by $x-1$.
- The quotient is $2x+2$ and remainder is 5, so $f(x)=2x+2 + \dfrac{5}{x-1}$.
- As $x \to \pm\infty$, the term $5/(x-1)$ tends to 0, so the graph approaches the line $y=2x+2$.
- Thus the oblique asymptote is $y=2x+2$.
- Answer: $\boxed{\text{Vertical asymptote: } x=1;\; \text{Oblique asymptote: } y=2x+2.}$
FAQs
Can functions have more than one vertical asymptote?
Yes. Rational functions often have multiple vertical asymptotes, one for each real root of the denominator that does not cancel.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.