Tangent Lines, Limits, and Continuity

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Tangent Lines, Limits, and Continuity

Tangent Line: A line that touches a curve at exactly one point

Limit: A point on graph that the function approaches/gets arbitrarily close to. There can be left and right hand limits denoted by a positive or negative sign after the value being approached in the limit equation. If the left and right hand limits do not agree, then the limit DNE.

The Precise Definition of a Limit states that if f is a function defined on an open interval that contains the number a, except possible at a itself, then the limit as x->a is L, and is written as below.

If every number ? > 0 there is a number ? > 0 such that - if 0 < Ix-aI < ? then If(x) - LI < ?

Limit Laws:

** Need to add limits as x->∞

Continuous: A function is continuous when it meets 3 conditions: (suppose c is any value)

1. f(c) is defined

2. The lim of f(x) exists

3. The lim of f(x) is c

Differential: Has exactly one tangent line for each point of the curve.

If function f is differentiable at point a then it is also continuous at a. However, function f is continuous at point a it does not mean that it is differentiable at point a.

Cusp/Corner: Continuous, but not differentiable.

Discontinuous: Not continuous and not differentiable

Vertical Tangent: Continuous and Differentiable