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Calculus ILimits and continuity

Limits at infinity, and the lines a graph approaches.

Follow a graph far to the left or right, handle square roots without losing a sign, and find its asymptotes. Then solve the class questions one step at a time.

Follow the ends · Step 1 of 8

Keep walking along the x-axis

−25−20−15−10−505101520250123
At x→∞{x\to\infty}, move right without stopping. At x→−∞{x\to-\infty}, move left without stopping. Infinity is a direction of travel, not a number to plug in. The output might settle at a number, grow without bound, or keep oscillating.
Try it: Choose a direction and slide farther out. What height does the dot approach?
Travel
Distance from zero2
f(2)=1+11+(2)2≈1.2000{\displaystyle f(2)} = {\displaystyle 1} + {\displaystyle \frac{1}{1+(2)^2}} \approx {\displaystyle 1.2000}

Both ends approach height 1. A horizontal asymptote is a line the graph approaches at an end.

lim⁡x→±∞f(x)=1⇒y=1{\displaystyle \lim_{x\to\pm\infty}f(x)} = {\displaystyle 1\quad} \Rightarrow {\displaystyle \quad y} = {\displaystyle 1}

A function can have zero, one or two horizontal asymptotes: at most one per end. It may cross them on the way.

Check yourself
What does x → −∞ say?

Need a hand with infinity or asymptotes?

Bring your practice questions. We work through them together until every type feels routine.

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