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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 *x* ⇒ *c* *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 *x*^{2}/x^{2})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 *x*^{2}/x^{2})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.