MAT 21A Lecture Notes - Lecture 4: List Of Trigonometric Identities
MAT 21A – Lecture 4 – One-Sided Limits
• Not only can limits be evaluated when x approaches a number c, from both
sides, but also either when x approaches c from the left and from the right only.
From both sides Left-hand sided limit Right-hand sided limit
• The superscript - idicates that is approachig c fro the left while the
superscript + eas that is approachig c fro the right.
• In general, we can say that if both the left-hand and right-sided limits both have
the same limit such that
and
, then
• Example: Let G(x) be a piecewise defined function. G(x) =
= 5
= -8
Explanation: Pick which of the two functions either 2x-7 or 3x-5 to use. Since x is
approaching -1 from the left, you must choose the function where x < -1, which
is 2x-7 giving 2(-1) + 7 = 5. Similarly, if x is approaching -1 from the right, x > -1,
thus 3(-1) – 5 = -8.
= 1
does not exist.
Explanation: The limit of G(x) as x approaches -1 does not exist because the left
and right hand sided limits are not equal to each other. That is to say 5 8.
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Document Summary
Right-hand sided limit: the superscript (cid:862)-(cid:863) i(cid:374)dicates that (cid:454) is approachi(cid:374)g c fro(cid:373) the left while the superscript (cid:862)+(cid:863) (cid:373)ea(cid:374)s that (cid:454) is approachi(cid:374)g c fro(cid:373) the right. In general, we can say that if both the left-hand and right-sided limits both have the same limit such that lim (cid:4666)(cid:4667)= and lim +(cid:4666)(cid:4667)=, then lim (cid:4666)(cid:4667): example: let g(x) be a piecewise defined function. G(x) = {(cid:884)+7, < (cid:883) (cid:885) 5 > (cid:883) lim (cid:2869)+(cid:4666)(cid:4667) = -8 lim (cid:2869) (cid:4666)(cid:4667) = 5. Explanation: pick which of the two functions either 2x-7 or 3x-5 to use. Since x is approaching -1 from the left, you must choose the function where x < -1, which is 2x-7 giving 2(-1) + 7 = 5. Similarly, if x is approaching -1 from the right, x > -1, thus 3(-1) 5 = -8. lim (cid:2870)(cid:4666)(cid:4667) = 1 lim (cid:2869)(cid:4666)(cid:4667) does not exist.
