7. Computing Limits

The Special Limits on the previous page may be combined with the following Limit Laws to compute many limits. They are justified based on the intuitive definition. The formal proofs will appear in the chapter on Precise Limits and Continuity.

a.2. Limit Laws

Let \(f(x)\) and \(g(x)\) be functions for which \(\lim\limits_{x\to a}f(x)=L\) and \(\lim\limits_{x\to a}g(x)=M\) where \(L\) and \(M\) are finite. Then:

Addition Law:           \(\lim\limits_{x\to a}(f(x)+g(x))=L+M\)

Subtraction Law:         \(\,\lim\limits_{x\to a}(f(x)-g(x))=L-M\)

Product Law:           \(\,\,\lim\limits_{x\to a}f(x)\,g(x)=LM\)

Quotient Law:           \(\lim\limits_{x\to a}\dfrac{f(x)}{g(x)}=\dfrac{L}{M}\),
provided \(g(x)\neq 0\) for all \(x\) and \(M\neq 0\).

Power Law:             \(\,\lim\limits_{x\to a}(g(x))^{f(x)}=M^L\),
provided \(g(x)\ge0\) for all \(x\), \(M\ge0\), \((g(x))^{f(x) }\) is defined for all \(x\) and \(M^L\) is defined.

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

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 intermediate powers and result are defined."
• "The limit of a continuous function of a function is the continuous function of the limit of the function."

The product, quotient and power laws include the special cases: \[\lim_{x\to a}(cf(x))=cL\qquad\qquad \lim_{x\to a}\dfrac{1}{f(x)}=\dfrac{1}{L}\] \[\lim_{x\to a}(f(x))^{\,p}=L^{\,p}\qquad\qquad \lim_{x\to a}b^{f(x)}=b^{\,L}\] where \(c\), \(p\) and \(b\gt0\) are constants. These say

• "The limit of a constant times a function is the constant times the limit of the function."
• "The limit of the reciprocal of a function is the reciprocal of the limit of the function, provided the denominator is non-zero."
• "The limit of a power of a function is the power of the limit of the function, provided the intermediate powers and result are defined."
• "The limit of a number raised to a function of powers is the number raised to the limit of the function."

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

Frequently, we do not introduce the letters \(L\) and \(M\). Then the Laws say:

• Constants pull out of limits: \[ \lim_{x\to a}c f(x) = c \lim_{x\to a} f(x) \]
• The limit of a sum is the sum of the limits: \[ \lim_{x\to a}(f(x)+g(x))=\lim_{x\to a} f(x)+\lim_{x\to a} g(x) \]
• The limit of a difference is the difference of the limits: \[ \lim_{x\to a}(f(x)-g(x))=\lim_{x\to a} f(x)-\lim_{x\to a} g(x) \]
• The limit of a product is the product of the limits: \[ \lim_{x\to a}(f(x)\,g(x))=\lim_{x\to a} f(x)\lim_{x\to a} g(x) \]
• The limit of a reciprocal is the reciprocal of the limit, provided the denominator is non-zero: \[ \lim_{x\to a}\dfrac{1}{g(x)}=\dfrac{1}{\displaystyle\lim_{x\to a} g(x)} \]
• The limit of a quotient is the quotient of the limits, provided the denominator is non-zero: \[ \lim_{x\to a}\dfrac{f(x)}{g(x)} =\dfrac{\displaystyle\lim_{x\to a} f(x)}{\displaystyle\lim_{x\to a} g(x)} \]
• Continuous functions pull out of limits: \[ \lim_{x\to a}p\big(f(x)\big)=p\big(\lim_{x\to a} f(x)\big) \]
• The limit of a power is the power of the limits, provided all the exponentials exist: \[ \lim_{x\to a}\big(f(x)\big)^{\small g(x)} =\big(\lim_{x\to a} f(x)\big)^{\displaystyle\small\lim_{x\to a} g(x)} \]

In summary, the Limit Laws basically say that if we can plug in \(x=a\) 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 into a form where the Limit Laws do apply, as explained on the next page.