Video summary

Исследование функции. Часть 4. Асимптоты графика функции

Main summary

Key takeaways

Educational

Main ideas / lessons from the video

  • The topic is studying functions, continuing from earlier points:
    1. Domain of the function
    2. Parity (even/odd behavior)
    3. Intersections with the coordinate axes
  • This video’s focus is the 4th point: asymptotes of the graph.

What asymptotes are (concept)

  • An asymptote is (informally, per the instructor’s wording) a straight line such that the distance from points on the curve to that line tends to 0 as the curve goes far away from the origin.
  • A “simpler” viewpoint: asymptotes are straight lines the graph approaches but (typically) does not cross.

Types of asymptotes (classification)

There are three types:

  1. Vertical asymptotes — lines of the form

    • (x=a)
    • Occur at discontinuities of the 2nd kind

  2. Horizontal asymptotes — lines of the form

    • (y=b)

  3. Inclined (slanted) asymptotes — lines of the form

    • (y=kx+b)

How graphs behave near asymptotes (intuition)

  • The graph may appear to “try” to intersect an asymptote, but usually cannot cross it.
  • In diagrams, asymptotes are often drawn as dotted lines (using different colors in the instructor’s examples).

Methodology / formulas for finding asymptotes (detailed instructions)

1) Vertical asymptotes

  • Find values (x=a) where the function has a discontinuity of the second kind.
  • For rational-type functions, vertical asymptotes occur where the denominator = 0 (and the function truly blows up).

Verification method:

  • Compute the one-sided limits as (x \to a^{-}) and (x \to a^{+}).
  • If both one-sided limits go to infinity (possibly with different signs), then (x=a) is a vertical asymptote.

Instructor’s criteria (as described): check left and right one-sided limits near (x=a); if both sides are infinite, it’s a discontinuity of the second kind → vertical asymptote.

2) Horizontal asymptote

  • Look for a limit of the form:
    • (y=b), where
      • (b=\lim_{x\to\infty} f(x))
  • If this limit equals a finite number, then a horizontal asymptote exists.
  • If the limit is infinite (e.g., (+\infty) or (-\infty)) or does not settle to a number as (x\to\infty), then no horizontal asymptote exists.

Practical computation approach mentioned:

  • If you get an indeterminate form like (\infty/\infty), divide numerator and denominator by the highest power of (x).

3) Inclined (slanted) asymptote

  • Assume an asymptote of the form:
    • (y=kx+b)

Steps:

  1. Compute the slope (k):
    • (k=\lim_{x\to\infty}\frac{f(x)}{x})
  2. Compute (b):
    • (b=\lim_{x\to\infty}\left(f(x)-kx\right))
  3. Substitute into:
    • (y=kx+b)

Worked example (as shown in the video, summarized)

The example function is:

  • [ f(x)=\frac{2x^2+x+10}{x-1} ]

Vertical asymptote

  • Denominator (x-1=0) ⇒ (x=1)
  • Verify using one-sided limits:
    • As (x\to 1^{-}): limit becomes (-\infty)
    • As (x\to 1^{+}): limit becomes (+\infty)

Conclusion: vertical asymptote (x=1).

Horizontal asymptote

  • Compute:
    • (b=\lim_{x\to\infty} f(x))
  • After reducing the (\infty/\infty) form, the limit does not approach a finite value (effectively behaves like infinity).

Conclusion: no horizontal asymptote.

Inclined asymptote

  1. Slope:
    • (k=\lim_{x\to\infty}\frac{f(x)}{x})
    • Result: (k=2)
  2. Intercept:
    • (b=\lim_{x\to\infty}(f(x)-2x))
    • Simplifies to a finite limit, giving (b=3)
  3. Therefore:
    • (y=2x+3)

Conclusion: inclined asymptote (y=2x+3).

Graph interpretation

  • The graph has:
    • vertical asymptote (x=1)
    • inclined asymptote (y=2x+3)
    • no horizontal asymptote
  • The instructor states the curve will approach these asymptotes and typically does not intersect them (noting special behavior may be covered later).

Additional notes on how many asymptotes a graph can have

  • A graph can have an infinite number of vertical asymptotes (e.g., behavior similar to tangent/cotangent).
  • There are no more than two horizontal asymptotes.
    • Often: one or two; none possible when the function doesn’t approach a finite constant.
  • There are usually no more than two inclined asymptotes (up to two may occur).

Speakers / sources featured

  • Ulyana Polovinkina — math tutor and the video narrator/instructor.

Original video