Some Examples for Limit ●Value of a function ●Area Problem OGiven a function f;find the area between the graph of fand an interval on the x-axis ●Tangent Line Problem OGive a function f;and a point on its graph,find an equation of the line that is tangent to the graph at the point
Some Examples for Limit ⚫Value of a function ⚫Area Problem Given a function f, find the area between the graph of f and an interval on the x-axis ⚫Tangent Line Problem Give a function f, and a point on its graph, find an equation of the line that is tangent to the graph at the point
Tangent Line
Tangent Line
Limit (An Informal View) If the value of f(x)can be made as close as we like to L by taking values of x sufficiently to a(not equal to a),then we write lim f(x)=L X->0 ●Read as: 0rf(x)>L,asx→a “the limit of f(x)as x approaches a is L.” ●Equivalently 1im(f(x)-L)=0
Limit (An Informal View) ⚫If the value of f (x) can be made as close as we like to L by taking values of x sufficiently to a (not equal to a), then we write ⚫Read as: “the limit of f (x) as x approaches a is L.” ⚫Equivalently lim ( ) ( ) , x a f x L or f x L as x a → = → → lim ( ) 0 ( ) x a f x L → − =
Examples of Limit Value x-1 f(x)= (N-1)(x+1) 4Sx→1， -1 0.999 0.9999 0.99999 1 1.00001 1.0001 1.001 X f(x) 1.999500 1.999950 1.999995 2 2.000005 2.000050 2.000500 Sinx asx→0， 8(x)= >? x -0.001 -0.0001 -0.00001 0/ 0.00001 0.0001 0.001 (x) 0.9999998 0.999999 1 g 0.9999999 0.999999 0.9999999 0.9999998 98 99998 99998 98
Examples of Limit Value ⚫ ⚫ 1 ( 1)( 1) 1, ( ) ? 1 1 x x x as x f x x x − − + → = = → − − x 0.999 0.9999 0.99999 1 1.00001 1.0001 1.001 f (x) 1.999500 1.999950 1.999995 2 2.000005 2.000050 2.000500 sin 0, ( ) ? x as x g x x → = → x -0.001 -0.0001 -0.00001 0 0.00001 0.0001 0.001 g (x) 0.9999998 0.9999999 98 0.999999 99998 1 0.999999 99998 0.9999999 98 0.9999998
Example of No Limit Value asx→0，f(x)=sin π x πk f(x) ±2 ±π/2 ±1 士1 士π 0 ±2/3 ±T/2 ±1 ±10/100 士10 0 ±10/105 ±(10+1/2)π ±1 ±10/1000 ±100元 0 ±10/1005 ±(100+1/2) ±1 ±10/10000 士1000元 0 ±10/10005 ±(1000+1/2)π ±1 ±10/100000 +10000元 0
Example of No Limit Value ⚫ ⚫ as x f x 0, ( ) sin ? x → = → x π/x f(x) ±2 ±π/2 ±1 ±1 ±π 0 ±2/3 ±π/2 ±1 ±10/100 ±10π 0 ±10/105 ±(10+1/2)π ±1 ±10/1000 ±100π 0 ±10/1005 ±(100+1/2)π ±1 ±10/10000 ±1000π 0 ±10/10005 ±(1000+1/2)π ±1 ±10/100000 ±10000π 0