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Limits and Continuity

This page is meant to serve as a quick overview of limits and continuity.

Limit

f has a limit L as x approaches c. Lim xc f(x) = L
Limit of any constant is a constant lim(x) ⇒ c(k) = k
Sum Rule lim(x) ⇒ c(x + 6) = lim x ⇌ c(x) + lim (x) ⇌ c(6) = c + 6
Difference Rule lim(x) ⇒ c(x - 6) = lim x ⇌ c(x) - lim (x) ⇌ c(6) = c - 6
Constant Multiple Rule lim(x) ⇒ c(5*x) = 5*lim x ⇌ c(x) = 5*c

Some limits can have a limit only from one side. lim(x) ⇒ c+ denotes from the right and lim(x) ⇒ c- denotes from the left.

Horizontal Asymptote

y = b is a horizontal asymptote if lim x ⇒ ∞ + = b or lim x ⇒ ∞ - = b
For Horizontal Asymptote's if degree on bottom is less than degree on bottom is less than degree on top, the Horizontal Aysmptote is y=0. Bottome and Top are the same (3 x2/x2)the Horizontal Aysmptote is the leading coefficients (3). Degree on top is higher than bottom then the asymptote is obligue.

Verticle Asymptote

x = a is a verticle asymptote if lim x ⇒ a + = + or - ∞ or lim x ⇒ ∞ - = b
For Horizontal Asymptote's if degree on bottom is less than degree on bottom is less than degree on top, the Horizontal Aysmptote is y=0. Bottome and Top are the same (3 x2/x2)the Horizontal Aysmptote is the leading coefficients (3). Degree on top is higher than bottom then the asymptote is obligue.

Continuity

A point is continuous at a point if it's in the domain if lim x ⇒ c f(x)=f(c), if f(c) is defined, and the limit exists.

Types of Discontinuity

Discontinuities are either removable or non-removable, and there are three types. Jump, infinite, and oscillating.

 Function Valuse of Special Angles Removable Discontinuity   Jump Discontinuity Infinite Discontinuity Oscillating Discontinuity

A continuous function is a function that is continuous at every point.

IROC-lim h ⇒ 0 (f(a+h)-f(a/h))

A normal line to a curve at a poin is the line perpendicular to the tangent at that point.