单变量微积分(03):Limits and Continuity
2015-03-21 21:38
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1. 极限
简单的极限,我们可以通过直接代入法求解,如:limx→3x2+xx+1=3
我们知道我们在利用极限求导数时:
limx→x0ΔfΔx=limx→x0f(x0+Δx)−f(x0)Δx
如果直接用代入法的话,会出现分母为0的情况。
2. 连续
连续的定义:
We say f(x) is continuous at x0 whenlimx→x0f(x)=f(x0)
四类不连续点
1. Removable Discontinuity
Right-hand limit: limx→x+0f(x) means limx→x0f(x) for x>x0.
Left-hand limit: limx→x−0f(x) means limx→x0f(x) for x<x0.
If limx→x+0f(x)=limx→x−0f(x) but this is not f(x0), or if f(x0) is undefined, we say the discontinuity is
removable.
比如说sinxx,x≠0。
2. Jump Discontinuity
limx→x+0f(x) for(x<x0) exists, and limx→x−0f(x) for(x>x0)also exists, but they are NOT equal.
3. Infinite Discontinuity
Right-hand limit: limx→0+1x=∞
Left-hand limit: limx→0−1x=−∞
4. Other(Ugly) discontinuity
This function doesn’t even go to±∞ — it doesn’t make sense to say it goes to anything. For something like this, we say the limit does not exist.
3. 两个三角函数的极限
注意下面的表达式中θ代表弧度,而不是角度。limθ→0sinθθ=1;
limθ→01−cosθθ=0;
几何证明:
当上图中的角度θ变得非常小的时候,我们可以看出半弦长(sinθ)越来越接近半弧长(θ)。
从上图中可以看出当角度变得越来越小时,1−cosθ相对于θ来显得越来越小。
4. 定理:可微则一定连续
If f is differentiable at x0, then f is continuous at x0.Proof:
limx→x0(f(x)−f(x0))=limx→x0[f(x)−f(x0)x−x0](x−x0)=f′(x0)⋅0=0
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