Defining Limits and Using Limit Notation - AP Calculus AB
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How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?
How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?
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$\lim_{{x \to a^-}} f(x)$. The $-$ superscript indicates approaching from values less than $a$.
$\lim_{{x \to a^-}} f(x)$. The $-$ superscript indicates approaching from values less than $a$.
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Identify the limit law for the constant multiple of a function.
Identify the limit law for the constant multiple of a function.
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$\lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x)$. Constants can be factored out of limit expressions.
$\lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x)$. Constants can be factored out of limit expressions.
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What is the limit of $f(x) = c$ as $x$ approaches any $a$?
What is the limit of $f(x) = c$ as $x$ approaches any $a$?
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$c$. Constant functions have limits equal to the constant value everywhere.
$c$. Constant functions have limits equal to the constant value everywhere.
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State the limit law for the product of two functions.
State the limit law for the product of two functions.
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$\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$. Limit of product equals product of limits when both individual limits exist.
$\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$. Limit of product equals product of limits when both individual limits exist.
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What does it mean if $\lim_{{x \to a}} f(x) = L$?
What does it mean if $\lim_{{x \to a}} f(x) = L$?
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As $x$ approaches $a$, $f(x)$ approaches $L$. The function value gets arbitrarily close to $L$ as input approaches $a$.
As $x$ approaches $a$, $f(x)$ approaches $L$. The function value gets arbitrarily close to $L$ as input approaches $a$.
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Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$.
Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$.
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- This is a fundamental trigonometric limit used in calculus.
- This is a fundamental trigonometric limit used in calculus.
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What is the condition for the existence of a two-sided limit?
What is the condition for the existence of a two-sided limit?
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Both one-sided limits must exist and be equal. When $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x)$, the two-sided limit exists.
Both one-sided limits must exist and be equal. When $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x)$, the two-sided limit exists.
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Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.
Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.
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$\infty$. As $x$ approaches $0^+$, $\frac{1}{x}$ becomes arbitrarily large and positive.
$\infty$. As $x$ approaches $0^+$, $\frac{1}{x}$ becomes arbitrarily large and positive.
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How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?
How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?
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$\lim_{{x \to a^+}} f(x)$. The $+$ superscript indicates approaching from values greater than $a$.
$\lim_{{x \to a^+}} f(x)$. The $+$ superscript indicates approaching from values greater than $a$.
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Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$.
Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$.
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- Divide numerator and denominator by $x^3$; limit of $\frac{3x^3}{x^3} = 3$.
- Divide numerator and denominator by $x^3$; limit of $\frac{3x^3}{x^3} = 3$.
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State the limit law for powers of a function.
State the limit law for powers of a function.
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$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$. Limit of power equals power of limit when the base limit exists.
$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$. Limit of power equals power of limit when the base limit exists.
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State the limit law for the difference of two functions.
State the limit law for the difference of two functions.
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$\lim_{{x \to a}} [f(x) - g(x)] = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)$. Limit of difference equals difference of limits when both individual limits exist.
$\lim_{{x \to a}} [f(x) - g(x)] = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)$. Limit of difference equals difference of limits when both individual limits exist.
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Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$.
Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$.
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- Direct substitution: $(-1)^3 + 2(-1)^2 + (-1) = -1 + 2 - 1 = 0$.
- Direct substitution: $(-1)^3 + 2(-1)^2 + (-1) = -1 + 2 - 1 = 0$.
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What is the definition of a limit in calculus?
What is the definition of a limit in calculus?
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The value a function approaches as the input approaches a point. This captures the fundamental concept of approaching a value without necessarily reaching it.
The value a function approaches as the input approaches a point. This captures the fundamental concept of approaching a value without necessarily reaching it.
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What is a one-sided limit?
What is a one-sided limit?
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The limit of a function as $x$ approaches a point from one side, either the left or the right. Considers approach from only left ($a^-$) or right ($a^+$) side of the point.
The limit of a function as $x$ approaches a point from one side, either the left or the right. Considers approach from only left ($a^-$) or right ($a^+$) side of the point.
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What is the limit notation for $f(x)$ as $x$ approaches infinity?
What is the limit notation for $f(x)$ as $x$ approaches infinity?
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$\lim_{{x \to \infty}} f(x)$. Examines function behavior as input values become arbitrarily large.
$\lim_{{x \to \infty}} f(x)$. Examines function behavior as input values become arbitrarily large.
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How is an infinite limit denoted?
How is an infinite limit denoted?
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$\lim_{x \to a} f(x) = \infty$ or $-\infty$. Uses infinity symbol to show unbounded growth in positive or negative direction.
$\lim_{x \to a} f(x) = \infty$ or $-\infty$. Uses infinity symbol to show unbounded growth in positive or negative direction.
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State the limit law for the quotient of two functions.
State the limit law for the quotient of two functions.
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$\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)}$, if $\lim_{{x \to a}} g(x) \neq 0$. Limit of quotient equals quotient of limits when denominator limit is nonzero.
$\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)}$, if $\lim_{{x \to a}} g(x) \neq 0$. Limit of quotient equals quotient of limits when denominator limit is nonzero.
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Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$.
Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$.
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$\frac{5}{2}$. Divide by highest power $x^2$: $\frac{5-\frac{4}{x}}{2+\frac{3}{x^2}} \to \frac{5}{2}$.
$\frac{5}{2}$. Divide by highest power $x^2$: $\frac{5-\frac{4}{x}}{2+\frac{3}{x^2}} \to \frac{5}{2}$.
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Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.
Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.
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$-\infty$. As $x$ approaches $0^-$, $\frac{1}{x}$ becomes arbitrarily large and negative.
$-\infty$. As $x$ approaches $0^-$, $\frac{1}{x}$ becomes arbitrarily large and negative.
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If $\lim_{{x \to a}} f(x) = L$, what is the relationship between $f(x)$ and $L$?
If $\lim_{{x \to a}} f(x) = L$, what is the relationship between $f(x)$ and $L$?
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As $x$ approaches $a$, $f(x)$ gets arbitrarily close to $L$. The function output approaches $L$ but doesn't necessarily equal $L$ at $x = a$.
As $x$ approaches $a$, $f(x)$ gets arbitrarily close to $L$. The function output approaches $L$ but doesn't necessarily equal $L$ at $x = a$.
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Evaluate $\lim_{{x \to 3}} (x^2 - 9)$.
Evaluate $\lim_{{x \to 3}} (x^2 - 9)$.
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- Direct substitution: $(3)^2 - 9 = 9 - 9 = 0$.
- Direct substitution: $(3)^2 - 9 = 9 - 9 = 0$.
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What does continuity at a point $x = a$ require?
What does continuity at a point $x = a$ require?
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$f(a)$ is defined, $\lim_{{x \to a}} f(x)$ exists, and $\lim_{{x \to a}} f(x) = f(a)$. Three conditions ensure no gaps, jumps, or undefined points.
$f(a)$ is defined, $\lim_{{x \to a}} f(x)$ exists, and $\lim_{{x \to a}} f(x) = f(a)$. Three conditions ensure no gaps, jumps, or undefined points.
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What is the limit of $f(x) = x$ as $x$ approaches $a$?
What is the limit of $f(x) = x$ as $x$ approaches $a$?
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$a$. The identity function $f(x) = x$ approaches the point value.
$a$. The identity function $f(x) = x$ approaches the point value.
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How is the limit of a function as $x$ approaches $a$ denoted?
How is the limit of a function as $x$ approaches $a$ denoted?
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$\lim_{{x \to a}} f(x)$. Standard mathematical notation using $\lim$ with subscript showing the approach.
$\lim_{{x \to a}} f(x)$. Standard mathematical notation using $\lim$ with subscript showing the approach.
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What is an infinite limit?
What is an infinite limit?
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A limit where $f(x)$ increases or decreases without bound as $x$ approaches a point. Function values grow without bound, approaching positive or negative infinity.
A limit where $f(x)$ increases or decreases without bound as $x$ approaches a point. Function values grow without bound, approaching positive or negative infinity.
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State the limit law for the sum of two functions.
State the limit law for the sum of two functions.
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$\lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)$. Limit of sum equals sum of limits when both individual limits exist.
$\lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)$. Limit of sum equals sum of limits when both individual limits exist.
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Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$.
Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$.
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- Factor as $\frac{(x+1)(x-1)}{x-1}$, cancel to get $x+1$, then substitute $x=1$.
- Factor as $\frac{(x+1)(x-1)}{x-1}$, cancel to get $x+1$, then substitute $x=1$.
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Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$.
Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$.
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$\infty$. Polynomial with positive leading coefficient grows without bound as $x \to \infty$.
$\infty$. Polynomial with positive leading coefficient grows without bound as $x \to \infty$.
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What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$?
What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$?
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Right-hand limit and left-hand limit, respectively. These describe approaches from the positive and negative sides respectively.
Right-hand limit and left-hand limit, respectively. These describe approaches from the positive and negative sides respectively.
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