Honors Precalculus Calculus Chapter 1 – Limits

Practice questions covering limits, continuity, horizontal and vertical asymptotes, and rates of change from Honors Precalculus Chapter 1.

What does the notation $\lim_{x \to c} f(x) = L$ specifically mean?

It means that the values of $f(x)$ approach or equal $L$ as the values of $x$ approach (but do not equal) $c$.

What are the four methods listed for finding the limits of function values?

Substitution, graphical investigation, numerical approximation, and algebra (or a combination of these).

What is the value of the trigonometric limit $\lim_{x \to 0} \frac{\sin(x)}{x}$?

$$1$$

What is the value of the trigonometric limit $\lim_{x \to 0} \frac{1 - \cos(x)}{x}$?

$$0$$

According to the notes, what are the three steps/techniques used to evaluate limits to resolve an indeterminate form of $\frac{0}{0}$?

1. Factoring, 2. FOIL, and 3. Rationalize.

How is the Horizontal Asymptote (HA) determined if the highest power of the function is in the denominator?

There is a horizontal asymptote at $y = 0$.

How is the Horizontal Asymptote (HA) determined if the highest power is the same in both the numerator and the denominator?

Divide the coefficients of these powers to find the asymptote.

What are the three requirements for a function $f(x)$ to be continuous at point $c$?

1. The limit must exist ($\lim_{x \to c} f(x)$), 2. the point must exist ($f(c)$), and 3. the limit must equal the point ($\lim_{x \to c} f(x) = f(c)$).

What are the four typical types of discontinuity?

Removable (hole), Jump, Infinite (vertical asymptote), and Oscillating.

How is a 'Removable' discontinuity defined?

A point where the limits from both sides exist and are equal, but the point does not match the limit or is undefined.

What does the Intermediate Value Theorem (IVT) state regarding continuous functions?

It guarantees that a function continuous on an interval will hit every value from $f(a)$ to $f(b)$, meaning the curve is unbroken with no jumps or separate branches.

What is the graphical interpretation of the Average Rate of Change (AROC)?

The slope of a secant line to a curve.

How is the Instantaneous Rate of Change (IROC) at any time $t$ defined?

$$\lim_{h \to 0} \frac{f(t+h) - f(t)}{h}$$

What is a Normal Line in relation to a tangent line?

A normal line is perpendicular to the tangent at a point, and its slope is the negative reciprocal of the slope of the tangent line.

Rank the following functions by 'strength' from weakest to strongest for end behavior modeling: exponential, polynomial, logarithmic, constant.

\text{constant} < \text{logarithmic} < \text{polynomial} < \text{exponential}

What is the difference between velocity and speed according to the physics application in the notes?

Velocity is a vector and can be positive or negative, while speed is not a vector and is only positive.

Under what name is the Squeeze Theorem also referred to in Section 1.1?

Sandwich Theorem.