Connecting Limits at Infinity and Horizontal Asymptotes · 连接无穷远处的极限与水平渐近线
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| end behavior/end bɪˈheɪvjə/ | 末端行为 | mò duān xíng wéi |
| limits at infinity/ˈlɪmɪts æt ɪnˈfɪnɪti/ | 无穷远处的极限 | wú qióng yuǎn chù de jí xiàn |
| horizontal asymptote/ˌhɒrɪˈzɒntl ˈæsɪmptəʊt/ | 水平渐近线 | shuǐ píng jiàn jìn xiàn |
Zoom out: what happens far away?
- Instead of $x\to c$, now let $x$ run off to $+\infty$ or $-\infty$.
- This describes the function's end behavior 末端行为 — its shape at the far left and far right.
- We write $\displaystyle\lim_{x\to\infty}f(x)=L$: as $x$ grows huge, $f$ settles toward $L$.
- These are limits at infinity 无穷远处的极限, and they reveal horizontal trends.
拉远看:很远处发生了什么?
- 不再是 $x\to c$,现在让 $x$ 跑向 $+\infty$ 或 $-\infty$。
- 这描述函数的末端行为——它在最左和最右处的形状。
- 我们写 $\displaystyle\lim_{x\to\infty}f(x)=L$:当 $x$ 变得极大,$f$ 稳定趋向 $L$。
- 这些是无穷远处的极限,它们揭示水平趋势。
Race of the highest powers
- For a rational function, end behavior is a race between the top and bottom degrees.
- Bottom wins (denominator higher degree): $f\to0$. Example $\dfrac{x}{x^2+1}\to0$.
- Tie (equal degrees): $f\to$ the ratio of leading coefficients. Example $\dfrac{3x^2+1}{x^2-5}\to3$.
- Top wins (numerator higher degree): $f\to\pm\infty$ — no horizontal asymptote (it grows).
最高次幂的赛跑
- 对有理函数,末端行为是上下最高次数之间的一场赛跑。
- 分母赢(分母次数更高):$f\to0$。例 $\dfrac{x}{x^2+1}\to0$。
- 平局(次数相等):$f\to$ 首项系数之比。例 $\dfrac{3x^2+1}{x^2-5}\to3$。
- 分子赢(分子次数更高):$f\to\pm\infty$——没有水平渐近线(它增长)。
A curve that flattens far out · 一条在远处变平缓的曲线
y = a / (x − b) + d
Slide $x$ toward the edges — a reciprocal-type curve flattens toward a horizontal asymptote as $x\to\pm\infty$. · 将$x$移向边缘——一种倒数型曲线在$x\to\pm\infty$处趋于变平，接近一条水平渐近线。
What is $\displaystyle\lim_{x\to\infty}\dfrac{x}{x^2+1}$? · $\displaystyle\lim_{x\to\infty}\dfrac{x}{x^2+1}$是什么？
Denominator degree is higher, so the fraction shrinks to $0$. · 分母次数更高，因此分数收缩至$0$。
Select all · 所有 rational functions with horizontal asymptote $y=0$. · 选择所有具有水平渐近线$y=0$的有理函数。
Denominator degree higher → $y=0$. The third ties in degree, giving $y=3$ instead. · 分母次数更高 → $y=0$。第三个在次数上打平，给出$y=3$而不是前者。
The horizontal asymptote
- If $\displaystyle\lim_{x\to\pm\infty}f(x)=L$ (a finite number), the line $y=L$ is a horizontal asymptote 水平渐近线.
- The graph flattens out and runs alongside $y=L$ far from the origin.
- Trick: divide every term by the highest power of $x$ in the denominator, then send $x\to\infty$ (each $\tfrac1{x^n}\to0$).
- $\dfrac{3x^2+1}{x^2-5}=\dfrac{3+\tfrac1{x^2}}{1-\tfrac5{x^2}}\to\dfrac{3}{1}=3$.
水平渐近线
- 若 $\displaystyle\lim_{x\to\pm\infty}f(x)=L$(一个有限数),那么直线 $y=L$ 是一条水平渐近线。
- 图像在远离原点处变平,与 $y=L$ 并行。
- 诀窍:每一项都除以分母中 $x$ 的最高次幂,再令 $x\to\infty$(每个 $\tfrac1{x^n}\to0$)。
- $\dfrac{3x^2+1}{x^2-5}=\dfrac{3+\tfrac1{x^2}}{1-\tfrac5{x^2}}\to\dfrac{3}{1}=3$。
Evaluate · 评价 $\displaystyle\lim_{x\to\infty}\dfrac{3x^2+1}{x^2-5}$. · 计算 $\displaystyle\lim_{x\to\infty}\dfrac{3x^2+1}{x^2-5}$。
Equal degrees → ratio of leading coefficients $\frac31=3$. · 次数相等 → 首项系数之比$\frac31=3$。
A finite limit at infinity, $\lim_{x\to\infty}f(x)=L$, gives a ____ asymptote $y=L$. · 无穷远处的有限极限$\lim_{x\to\infty}f(x)=L$给出了一条____渐近线$y=L$。
The graph flattens toward $y=L$. · 图形向$y=L$变平。
The two ends can differ
- A function may approach one value as $x\to+\infty$ and a different value as $x\to-\infty$.
- $\arctan x\to\tfrac{\pi}{2}$ on the right but $\to-\tfrac{\pi}{2}$ on the left — two horizontal asymptotes.
- So always check both directions; don't assume symmetry.
- Unlike a vertical asymptote (a break in the middle), a horizontal asymptote is about the far-away trend, and a curve may even cross it.
两端可以不同
- 一个函数在 $x\to+\infty$ 时可能趋向一个值,而在 $x\to-\infty$ 时趋向另一个值。
- $\arctan x$ 右侧 $\to\tfrac{\pi}{2}$,左侧 $\to-\tfrac{\pi}{2}$——两条水平渐近线。
- 所以永远检查两个方向;别假设对称。
- 与竖直渐近线(中间的断裂)不同,水平渐近线说的是远处的趋势,曲线甚至可以穿过它。
A graph can cross its horizontal asymptote in the middle and still approach it at the ends. · 图形可以在中间穿过其水平渐近线，同时仍趋向于它在两端。
Horizontal asymptotes describe far-away behavior only; crossing in the middle is allowed. · 水平渐近线仅描述远处的行为；中间穿过是允许的。
For · 支持 $\arctan x$, the two limits $\lim_{x\to+\infty}$ and $\lim_{x\to-\infty}$ are... · 对于$\arctan x$，两个极限$\lim_{x\to+\infty}$和$\lim_{x\to-\infty}$是...
The two ends can differ — here two horizontal asymptotes. · 两端可以不同——这里有两个水平渐近线。
A vertical asymptote and a horizontal asymptote are different beasts. Vertical ($x=c$): the output blows up at a specific input — a graph never crosses it. Horizontal ($y=L$): the far-away trend as $x\to\pm\infty$ — a graph may cross it in the middle and still approach it at the ends.
竖直渐近线和水平渐近线是两种不同的东西。竖直($x=c$):输出在某个特定输入处爆发——图像永不穿过它。水平($y=L$):$x\to\pm\infty$ 时的远处趋势——图像在中间可以穿过它,却仍在两端趋近它。
Find $\displaystyle\lim_{x\to\infty}\dfrac{2x^2-3x}{5x^2+7}$ and the horizontal asymptote.
- Degrees tie (both $2$), so divide by $x^2$: $\dfrac{2-\tfrac3x}{5+\tfrac7{x^2}}$.
- As $x\to\infty$, the $\tfrac3x$ and $\tfrac7{x^2}$ terms $\to0$.
- Limit $=\dfrac{2}{5}$, so $y=\tfrac25$ is the horizontal asymptote.
求 $\displaystyle\lim_{x\to\infty}\dfrac{2x^2-3x}{5x^2+7}$ 及水平渐近线。
- 次数平局(都是 $2$),所以除以 $x^2$:$\dfrac{2-\tfrac3x}{5+\tfrac7{x^2}}$。
- 当 $x\to\infty$,$\tfrac3x$ 和 $\tfrac7{x^2}$ 项 $\to0$。
- 极限 $=\dfrac{2}{5}$,所以 $y=\tfrac25$ 是水平渐近线。
Limits at infinity describe end behavior. For rationals: denominator wins → $0$; degrees tie → ratio of leading coefficients; numerator wins → $\pm\infty$ (no horizontal asymptote). A finite $\lim_{x\to\pm\infty}f(x)=L$ gives a horizontal asymptote $y=L$, and the two ends may differ.
无穷远处的极限描述末端行为。对有理函数:分母赢 → $0$;次数平局 → 首项系数之比;分子赢 → $\pm\infty$(无水平渐近线)。有限的 $\lim_{x\to\pm\infty}f(x)=L$ 给出一条水平渐近线 $y=L$,而两端可以不同。