Topic 1.5 Determining Limits Using Algebraic Properties of Limits: apply the limit laws (sum, difference, product, quotient, constant multiple, power) and direct substitution to evaluate limits.

What this topic is asking

The College Board (Topic 1.5) wants you to evaluate limits using the limit laws and direct substitution. When a function is built from sums, products, quotients and powers of pieces whose limits you know, the limit of the whole is built the same way from the limits of the pieces. For most "nice" (continuous) functions, this just means substituting the value of $a$.

The limit laws

These let you decompose a complicated limit into simple ones. The basic building blocks are $\lim_{x \to a} c = c$ (a constant) and $\lim_{x \to a} x = a$ (the identity).

Direct substitution

So $\lim_{x \to 2} (x^2 + 3x - 1) = 2^2 + 3(2) - 1 = 9$ just by plugging in, because polynomials are continuous everywhere.

Reading the result of substitution

After substituting, you see one of three things:

• A real number: that is the limit. Done.
• $\frac{\text{nonzero}}{0}$: the limit is infinite (does not exist as a finite value); check signs for $\pm\infty$.
• $\frac{0}{0}$: an indeterminate form. The limit may still exist, but you must simplify first (factor, rationalize, or use another method) - this is Topic 1.6.

Using the laws with unknown limits

The limit laws are not only for plugging in numbers; they are also how you combine limits you are told but cannot compute, which is a frequent exam format. If a problem states $\lim_{x \to a} f(x) = 4$ and $\lim_{x \to a} g(x) = -2$ without giving formulas, you still find $\lim_{x \to a}[f(x) + 3g(x)]$, $\lim_{x \to a}[f(x)g(x)]$, or $\lim_{x \to a}\sqrt{f(x)}$ purely by applying the corresponding law to the given values. Here the laws are doing real work, because no substitution is possible. The one law to apply with care is the quotient law: before writing $\frac{\lim f}{\lim g}$ you must confirm $\lim g \neq 0$, since a zero denominator-limit means the quotient law does not apply and the limit needs separate analysis. This "given the parts, build the whole" skill is exactly what the abstract limit-law questions test.

Why substitution can fail

Direct substitution works precisely when the function is continuous at $a$. If $a$ is not in the domain (for instance it makes a denominator zero), substitution will not give a clean number, and the algebraic-manipulation methods of the next topic take over. The discipline is always the same: substitute first, then read what kind of result you got. Because the standard families - polynomials, rationals on their domains, roots, trig, exponential and logarithmic functions - are continuous wherever they are defined, substitution succeeds for the large majority of limits you meet, and only the special $\frac{0}{0}$ and $\frac{k}{0}$ cases need extra work. Knowing that substitution is the default, not a lucky shortcut, keeps your method orderly.