Limits and Continuity Flash Cards

The limit @@\lim_{x\rightarrow b}f\left(x\right)@@ exist

• @@\lim_{x\rightarrow b^-}f\left(x\right)=\lim_{x\rightarrow b^+}f\left(x\right)@@, the left and right sided limit exist and are equal to each other

• @@\lim_{x\rightarrow b^-}f\left(x\right)\neq\lim_{x\rightarrow b^+}f\left(x\right)@@, the left and right sided limit do not exist

• @@\lim_{x\rightarrow b}f\left(x\right)=0@@, the limit is equal to zero

• @@\lim_{x\rightarrow b}f\left(x\right)\text{ does not exist}@@, the limit is undefined

• @@\frac{f\left(b\right)-f\left(a\right)}{b-a}@@

• @@f\left(b\right) + f\left(a\right)@@

• @@\frac{f\left(a\right)}{b}@@

• @@f\left(b\right) - f\left(a\right)@@

Type of discontinuity

• Removable

• Jump

• Infinite

• Essential

Discontinuity @@f\left(x\right)=\frac{\left|x-a\right|}{x-a}@@

• Removable Discontinuity at @@x=a@@

• Infinite Discontinuity at @@x=a@@

• Oscillating Discontinuity at @@x=a@@

The function @@f\left(x\right)@@ is continuous at @@x=b@@

• @@f\left(b\right)@@ is defined

• @@\lim_{x\rightarrow b^-}f\left(x\right) \neq \lim_{x\rightarrow b^+}f\left(x\right)@@

• @@\lim_{x\rightarrow b}f\left(x\right) \text{ does not exist}@@

• @@\lim_{x\rightarrow b}f\left(x\right) \neq f\left(b\right)@@