Top 10 Things You Need to Know About Limits

A comprehensive set of practice flashcards covering fundamental limit concepts, calculation techniques, indeterminate forms, limit laws, L'Hopital's rule, the Squeeze Theorem, and the formal epsilon-delta definition based on the lecture.

What condition must be met for the overall limit $\lim_{x \to a} f(x)$ to equal $l$?

The y-values of the function must approach $l$ from both the left side ($x \to a^-$) and the right side ($x \to a^+$) of $a$.

In limit notation, what do the minus and plus superscripts represent in $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$?

The minus sign indicates $x$ is approaching $a$ from the left side of $a$, and the plus sign indicates $x$ is approaching $a$ from the right side of $a$.

In the Chapter 1 piecewise function example, why does the overall limit $\lim_{x \to 1} f(x)$ fail to exist?

The right-hand limit as $x \to 1^+$ is $2$ and the left-hand limit as $x \to 1^-$ is $0$. Since the left and right limits are not equal, the overall limit does not exist.

What is the first technique to attempt when evaluating any limit from an equation?

Direct substitution, which involves plugging the target value $a$ directly into the function for $x$.

What three outcomes can result from direct substitution when evaluating a limit as $x \to a$?

1. A finite number $f(a)$, which is the limit value. 2. The indeterminate form $\frac{0}{0}$. 3. A nonzero number divided by zero, which indicates the limit does not exist.

What does a result of a nonzero number divided by zero during direct substitution typically indicate about the graph at $x = a$?

It indicates that the limit does not exist and that there is typically a vertical asymptote at $x = a$.

How is the limit $\lim_{x \to 1} (x^2 - 4)$ evaluated using direct substitution?

Substitute $x = 1$ directly into the expression to obtain $1^2 - 4 = -3$.

When evaluating $\lim_{x \to -3} \frac{2x + 5}{x + 3}$, what are the values of the left-hand and right-hand limits?

Direct substitution yields $\frac{-1}{0}$. The left-hand limit as $x \to -3^-$ is $\infty$, and the right-hand limit as $x \to -3^+$ is $-\infty$.

What rule justifies rewriting a function $f(x)$ as an equivalent function $g(x)$ when encountering an indeterminate form $\frac{0}{0}$?

If $f(x) = g(x)$ for all $x \neq a$, then $\lim_{x \to a} f(x) = \lim_{x \to a} g(x)$.

How do you evaluate $\lim_{x \to 5} \frac{x^2 - 7x + 10}{x - 5}$ using factoring?

Factor the numerator as $(x - 5)(x - 2)$, cancel the $(x - 5)$ terms to get $\lim_{x \to 5} (x - 2)$, and use direct substitution to get $5 - 2 = 3$.

How do you evaluate $\lim_{x \to 9} \frac{x - 9}{\sqrt{x} - 3}$ using a conjugate?

Multiply the top and bottom by $\sqrt{x} + 3$ to get $\frac{(x - 9)(\sqrt{x} + 3)}{x - 9}$, cancel $(x - 9)$, and evaluate $\sqrt{9} + 3 = 6$.

How do you evaluate $\lim_{x \to 0} \frac{\sin(2x)}{2\sin(x)}$ using a trig identity?

Substitute $\sin(2x) = 2\sin(x)\cos(x)$, cancel $\sin(x)$ and $2$, and evaluate $\lim_{x \to 0} \cos(x) = 1$.

What is the Limit Law for a constant multiple $\lim_{x \to a} [c \cdot f(x)]$?

$$c \cdot \lim_{x \to a} f(x)$$

How is a horizontal asymptote $y = b$ defined using limits?

A line $y = b$ is a horizontal asymptote of $f(x)$ if $\lim_{x \to \infty} f(x) = b$ or $\lim_{x \to -\infty} f(x) = b$.

What method is used to evaluate an infinite limit of a rational function that yields $\frac{\infty}{\infty}$?

Divide every term in the numerator and denominator by the highest power of $x$ present in the denominator.

How do you evaluate $\lim_{x \to \infty} \frac{5x^2 + 8x - 3}{3x^2 + 2}$?

Divide each term by $x^2$ to get $\lim_{x \to \infty} \frac{5 + \frac{8}{x} - \frac{3}{x^2}}{3 + \frac{2}{x^2}} = \frac{5 + 0 - 0}{3 + 0} = \frac{5}{3}$.

What is L'Hopital's rule for a quotient $\lim_{x \to c} \frac{f(x)}{g(x)}$ resulting in $\frac{0}{0}$ or $\frac{\infty}{\infty}$?

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$

How do you evaluate $\lim_{x \to 2} \frac{x^2 - 7x + 10}{x - 2}$ using L'Hopital's rule?

Differentiate numerator and denominator to get $\lim_{x \to 2} \frac{2x - 7}{1}$, then substitute $x = 2$ to obtain $2(2) - 7 = -3$.

How do you convert an indeterminate product of type $0 \times \infty$, such as $\lim_{x \to 0^+} (x \ln(x))$, to use L'Hopital's rule?

Rewrite the product as a quotient $\frac{\ln(x)}{\frac{1}{x}}$ to produce the form $\frac{-\infty}{\infty}$, then apply L'Hopital's rule.

How do you resolve an indeterminate difference of type $\infty - \infty$ to enable L'Hopital's rule?

Combine the terms into a single fraction using a common denominator to transform the expression into $\frac{0}{0}$, then apply L'Hopital's rule.

What three statements define the criteria and conclusion of the Squeeze Theorem?

1. $g(x)$ is trapped between $f(x)$ and $h(x)$ near $a$ ($f(x) \le g(x) \le h(x)$). 2. $\lim_{x \to a} f(x) = l$ and $\lim_{x \to a} h(x) = l$. 3. Therefore, $\lim_{x \to a} g(x) = l$.

How do you evaluate $\lim_{x \to 0} \left(x^2 \sin\left(\frac{1}{x}\right)\right)$ using the Squeeze Theorem?

Start with $-1 \le \sin\left(\frac{1}{x}\right) \le 1$, multiply by $x^2$ to get $-x^2 \le x^2 \sin\left(\frac{1}{x}\right) \le x^2$, and since $\lim_{x \to 0} (-x^2) = \lim_{x \to 0} (x^2) = 0$, the limit is $0$.

What is the formal Epsilon-Delta definition of a limit $\lim_{x \to a} f(x) = l$?

For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - l| < \epsilon$.

In the Epsilon-Delta definition of a limit, what do $|x - a|$ and $|f(x) - l|$ represent geometrically?

$|x - a|$ represents the distance from $x$ to $a$, and $|f(x) - l|$ represents the distance from $f(x)$ to $l$.