Understanding Infinite Limits in Calculus
Explore the concept of infinite limits in calculus, which allow for describing the behavior of functions as values become arbitrarily large. Learn about vertical asymptotes, dominant terms, and examples illustrating infinite limits. Dive into finding horizontal and vertical asymptotes of functions and understand their significance in function behavior analysis.
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Presentation Transcript
Infinite Limits Limits Involving
In this section we will extend the concept of limit to allow for infinite limits, which are not really limits at all but provide useful symbolism for describing the behavior of functions whose values become arbitrarily large, positive or negative. Infinite limits provide useful symbols and language for describing the behavior of functions whose values become arbitrarily large, positive or negative. We continue our analysis of graphs of rational functions using vertical asymptotes and dominant terms for numerically large values of ? .
? ?= . In writing this, we 1. We write lim are not saying that the limit exists. Nor are we saying that there is a real number , for there is no such number. Rather, we are saying that lim exist because ? ?becomes arbitrarily large and positive as ? +. ? +?(?) = lim ? + ? ?does not ? + ? ?= . Again, we are 2. We write lim not saying that the limit exists and equals the number . There is no real number . We are describing the behavior of a function whose limit as ? does not exist because its values become arbitrarily large and negative. ? ?(?) = lim ?
? ? Example: Find lim ? ?and lim ? ?. ? ? ? ?+ Solution: Think about the number ? ? and its reciprocal. As ? ?+, we have (? ?) ?+and ? ? ? . As ? ? , we have (? ?) ? and ? ? ? . Definition: A line ? = ? is a vertical asymptote of the graph of a function ?(?) if either lim lim ? ?+?(?) = . ? ? ?(?) = or
Example: Find the horizontal and vertical asymptotes of the curve ? ? =?+? ?+?. Solution: We are interested in the behavior as ? and as ? ? where the denominator is zero. Since ?+(? ?) ? ? ?+? ?+?= lim lim ? = ? ? ?+ and ?+? ?+?= ?+? ?+?= and lim ? ?+ lim ? ?+ then: the horizontal asymptote is ? = ? and the vertical asymptote is ? = ?.
Example: Find the horizontal and vertical asymptotes of the curve ? ? = ?? ? Solution: We are interested in the behavior as ? and as ? ? where the denominator is zero. Since ? ?? ? the line ? = ? is a horizontal asymptote of the graph of ? . Since ? ?? ?= and lim ? ? and ? ?? ?= and lim ? ? then the lines ? = ? and ? = ? are vertical asymptotes of the graph of ? . ? lim ? ? ?? ?= lim ? ?+ ? ?? ?= lim ? ?+
Limits Involving A central fact about (sin?)/? is that in radian measure its limit as ? ? is ? . We can see this in the figure below and confirm it algebraically using the Squeezing Theorem. ? ?sin ? Theorem:lim = ? where ? in radians. ? ? ?tan ? Corollary: lim = ? where ? ? in radians.
Corollary: sin?? ?? tan?? ?? tan ?? sin ??= lim ?? sin?? sin??=? tan?? tan??=? 1 lim ? 0 = lim ? 0 sin??= lim ?? tan??= lim sin ?? tan ??=? ? ? 0 ? lim ? 0 = lim ? 0 ? ? 0 ? lim ? 0 ? ? 0 tan 5? sin 3?. Example: Evaluate lim ? 0 tan 5? sin 3?=5 Solution: lim 3 ? 0
?cos ? ? Example: Show that lim = ?. ? ? Solution: Using the half-angle formula ???? = ? ?sin (? ?) we calculate ? [? ?sin (? ?)] ? cos? ? lim ? ? = lim ? ? ? ? ?[? sin(?/?) = lim sin(?/?)] ? (? ?) ? lim ? ?sin(? ?) = ? ?/? = ? lim ? ? = ? ? ?
? sin ??+tan ?? ??+?? Example: Evaluate lim . ? ? Solution: ?sin?? +tan?? ?sin?? + tan?? ?? + ?? ? ?? ?+?? ? lim ? ? = =lim ? ? ? ?????? +????? ? ? = lim ? ? ? + ? ? ? ?+? ? + ? =?? ? = ?
sin ?2 ? Example: Evaluate lim . ? ? Solution: sin?2 ? sin?2 ? ? ? = ? sin?2 ? lim ? ? = lim[ ? ? ?] = lim lim ? ?? ? ? sin ? ?? . Example: Evaluate lim ? ? Solution: 2 sin ? ?? =? ? ?[sin ? ??] =? =? sin ? ?? lim ? ? ?lim ?lim ? ? ? ?2=? ?
tan(??) ?? ??. Example: Evaluate lim ? ? Solution: tan(? ?) ?? ?? tan(? ?) (? ?)(? + ?) lim ? ? = lim ? ? tan(? ?) ? ? ? = lim ? ? lim ? ? ? + ? ? ? = ? ? + ?= ??
