Calc. 1( Limit of a Function and Intro Derivatives ( Tangent+ Velocity Problem

f(x+h)-f(x)/h

1. f(x) approaches a different number from the right side of a than it approaches from the left side.

2. f(x) increases or decreases without bound as x approaches a.

3. f(x) oscillates between two fixed values as x approaches a.

Lim x➡️a f(x)-f(a)/x-a

Lim h➡️0 f(a+h)-f(a)/h

Lim h➡️0 f(x+h)-f(x)/h

The limit of the right and left are the same value so the limit is same value.

You can distribute your limit to each part of the expression( put the limit before each factor)

The constant can be moved outside the limit and away from sitting next to the limit.

You can multiply each function by each other as long as they have the same limit

The value of the limit on num. and the limit of the denom. Equals the limit of the fun in num/ denom. As long as the denom. Does not equal zero.

As long as the n(interfere) is positive than the limit can be moved ext to then function and done together

If the upper function( right) and lower function(left) are equal and the function is between the other two , than that middle function equals the equaled value of the outside functions

If you can fill in the circle and create a continuous function( has a function value and limit)

Where you have a gap( break) between the line and you would have to go up or down to continue

An asymptote is usually a good indicator, but it’s when both the function value and limit does not exist in that limit

Be sure to write and not include anything discontinuous

Polynomials are continuous everywhere(-,+) Rational, root, trig, inverse trig, exponential, logarithmic functions are continuous where defined( every number) in its domain

Condition 1: Must be Continuous. 2: N must be between the intervals. 3: That function values don’t equal. To prove the function continuous at the interval.

Y=Mx+b ( just find value of m through formulas and solve for b using the points for x and y.

S(t)= S( t2)-S(t1)/ t2-t1 : Distance/ time

S’(t)=v(t)

Go to page 36( second page of 2.8) study how you make those graphs

If f is differentiable at a, then f is continuous at a. NOT! The other way around