18. Sequences

The Special Limits on the previous page may be combined with the following Limit Laws to compute many limits. The Limit Laws are intuitively obvious, but the proof links go to the proofs which are at the end of this chapter after the precise definition.

b4. Limit Laws

Let \(a_n\) and \(b_n\) be sequences for which \(\lim\limits_{n\to\infty}a_n=L\) and \(\lim\limits_{n\to\infty}b_n=M\) where \(L\) and \(M\) are finite. Then:

Addition Law:           \(\lim\limits_{n\to\infty}(a_n+b_n)=L+M\)
Proof

Subtraction Law:         \(\,\lim\limits_{n\to\infty}(a_n-b_n)=L-M\)
Proof

Product Law:           \(\,\,\lim\limits_{n\to\infty}a_n\,b_n=LM\)
Proof

Quotient Law:           \(\lim\limits_{n\to\infty}\dfrac{a_n}{b_n}=\dfrac{L}{M}\),
provided \(b_n\neq 0\) for all \(n\) and \(M\neq 0\).
Proof

Power Law:             \(\,\lim\limits_{n\to\infty}(b_n)^{a_n}=M^L\),
provided \(b_n\ge0\) for all \(n\), \(M\ge0\), \((b_n)^{a_n }\) is defined for all \(n\) and \(M^L\) is defined.
Proof

Continuous Function Law:   \(\lim\limits_{n\to\infty}p(a_n) =p(L)\),
where \(p(x)\) is any function which is continuous at \(L\).
Proof

In words, these say

• "The limit of a sum is the sum of the limits."
• "The limit of a difference is the difference of the limits."
• "The limit of a product is the product of the limits."
• "The limit of a quotient is the quotient of the limits, provided the denominator is non-zero."
• "The limit of a power is the power of the limits, provided the terms and result are defined."
• "The limit of a continuous function of a sequence is the function of the limit of the sequence."

The product, quotient and power laws include the special cases: \[\lim_{n\to\infty}(c\,a_n)=cL\qquad\qquad \lim_{n\to\infty}\dfrac{1}{a_n}=\dfrac{1}{L}\] \[\lim_{n\to\infty}(a_n)^{\,p}=L^{\,p}\qquad\qquad \lim_{n\to\infty}b^{\,a_n}=b^{\,L}\] where \(c\), \(p\) and \(b\) are constants. These say

• "The limit of a constant times a sequence is the constant times the limit of the sequence."
• "The limit of the reciprocal of a sequence is the reciprocal of the limit of the sequence."
• "The limit of a power of a sequence is the power of the limit of the sequence."
• "The limit of a number raised to a sequence of powers is the number raised to the limit of the sequence."

In general, to compute a limit, we repeatedly apply the Limit Laws until we get down to the Special Limits.

In summary, the Limit Laws basically say that if we can "plug in \(n=\infty\)" and get a finite number, \(L\), while satisfying all the conditions in the Limit Laws, then \(L\) is the limit.

All is very nice when the Limit Laws apply, but most of our time will be spent on limits where the Limit Laws do not apply. In that case, we first need to algebraically manipulate the limit before applying the Limit Laws, as explained on the next page.