Here is a handout on limit rules: 

Calculus I                            Carrie Diaz Eaton, Unity College

Special limits of sequences rules

1. When the base changes:

Discrete For an=n-k=1nk, k>0, thennan=0

Continuous or f(x) = x-k, k>0, then xf(x)=0

Example:

2. When the exponent changes:

    For an=a0n, or f(t)=x0t

1. if -1<<1, then nan=0 or tf(t)=0

        Example:

2. if >1, then nan DNE or tf(t) DNE

        Example:

3. Rational expressions:

        For an=bpnp+bp-1np-1+...+b1n+b0cqnq+cq-1nq-1+...+c1n+c0,

1. if p<q, then nan=0

    Example:

2. if p=q, then nan=bpcq

    Example:

3. if p>q and both p and q are positive or negative, then nan= (DNE)

    Example:

4. if p>q and only one of p and q are positive, then nan=- (DNE)

    Example:

4. Sandwich Principle

   1. General case:

    If  nan=ncn=L     and anbncn, then nbn=L

    Example:

2. Special case, Alternating sequences with (-1)n:

        If nan=0,     then nan=0,     otherwise the limit DNE

        Example: