Calculus: Limits and Continuity Practice Flashcards

Flashcards covering the concepts of limits, average and instantaneous rates of change, limit laws, one-sided limits, and precise definitions based on lecture notes.

What specific interests of seventeenth-century mathematicians motivated the study of limits?

The study of motion for objects near the earth and the motion of planets and stars, involving speed, direction, and tangent lines to paths of motion.

According to Galileo's law, what is the formula for the distance $y$ (in feet) fallen by a solid object after $t$ seconds?

$$y = 16t^2$$

How is the average speed of a moving object during a time interval calculated?

By dividing the distance covered ($\Delta y$) by the time elapsed ($\Delta t$).

What is the geometric interpretation of the average rate of change of a function $y = f(x)$ over the interval $[x_1, x_2]$?

The slope of the secant line passing through the points $P(x_1, f(x_1))$ and $Q(x_2, f(x_2))$.

What is the definition of the average rate of change of $y = f(x)$ with respect to $x$ over $[x_1, x_2]$?

$\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f(x_1 + h) - f(x_1)}{h}$, where $h \neq 0$.

How is the slope of a curve at a point $P$ defined?

The slope of the tangent line at $P$, which is the limit of the slopes of secant lines $PQ$ as $Q$ approaches $P$ along the curve.

What does it mean to say that $\lim_{x \to c} f(x) = L$ using the informal definition?

The values of $f(x)$ are arbitrarily close to the number $L$ whenever $x$ is sufficiently close to $c$ on either side of $c$, but not necessarily at $c$ itself.

What are three common reasons a function might fail to have a limit at a point?

1. The function jumps (e.g., unit step function). 2. The function grows too large in absolute value (not bounded). 3. The function oscillates too much (e.g., $\sin(1/x)$).

List the Sum, Difference, and Constant Multiple Rules for limits.

1. Sum Rule: $\lim_{x \to c} (f(x) + g(x)) = L + M$. 2. Difference Rule: $\lim_{x \to c} (f(x) - g(x)) = L - M$. 3. Constant Multiple Rule: $\lim_{x \to c} (k \times f(x)) = k \times L$.

Identify the Product and Quotient Rules for limits.

1. Product Rule: $\lim_{x \to c} (f(x) \times g(x)) = L \times M$. 2. Quotient Rule: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ (provided $M \neq 0$).

According to Theorem 2, how is the limit of a polynomial $P(x)$ at $x = c$ evaluated?

By direct substitution: $\lim_{x \to c} P(x) = P(c)$.

If the numerator and denominator of a rational function are both zero at $x = c$, what algebraic step is typically required to find the limit?

Factoring and canceling the common factor of $(x - c)$, then using substitution in the simplified fraction.

What is the Sandwich Theorem (also known as the Squeeze Theorem)?

If $g(x) \leq f(x) \leq h(x)$ for all $x$ in some open interval containing $c$ (except possibly at $x = c$) and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.

What is the precise definition of a limit?

Let $f(x)$ be defined on an open interval about $c$, except possibly at $c$. $\lim_{x \to c} f(x) = L$ if for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $x$, $0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$.

What is the relationship between one-sided limits and two-sided limits?

$f(x)$ has a limit as $x \to c$ if and only if both the left-hand limit ($\lim_{x \to c^-} f(x)$) and the right-hand limit ($\lim_{x \to c^+} f(x)$) exist and are equal.

What is the value of $\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta}$ when $\theta$ is measured in radians?

$$1$$

How is the right-hand limit $\lim_{x \to c^+} f(x) = L$ defined precisely?

For every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $x$, $c < x < c + \delta \implies |f(x) - L| < \epsilon$.