Estimating Limit Values from Graphs - AP Calculus AB
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How do you find $\lim_{x \to 5} f(x)$ if $f(x)$ is constant?
How do you find $\lim_{x \to 5} f(x)$ if $f(x)$ is constant?
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The constant value. Constant functions have the same value everywhere.
The constant value. Constant functions have the same value everywhere.
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What is the limit of $\lim_{x \to -a} f(x)$ for $f(x) = x^2$?
What is the limit of $\lim_{x \to -a} f(x)$ for $f(x) = x^2$?
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$a^2$. $(-a)^2 = a^2$ for any real number $a$.
$a^2$. $(-a)^2 = a^2$ for any real number $a$.
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What is $\lim_{x \to 0} \frac{1}{x}$?
What is $\lim_{x \to 0} \frac{1}{x}$?
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Does not exist; approaches $\infty$ as $x \to 0^+$ and $-\infty$ as $x \to 0^-$. Left and right limits differ, so the limit doesn't exist.
Does not exist; approaches $\infty$ as $x \to 0^+$ and $-\infty$ as $x \to 0^-$. Left and right limits differ, so the limit doesn't exist.
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What is the graphical significance of $\lim_{x \to c} f(x) = \infty$?
What is the graphical significance of $\lim_{x \to c} f(x) = \infty$?
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Vertical asymptote at $x = c$. Infinite limits create vertical asymptotes on graphs.
Vertical asymptote at $x = c$. Infinite limits create vertical asymptotes on graphs.
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What does $\lim_{x \to 0} e^{-x}$ approach?
What does $\lim_{x \to 0} e^{-x}$ approach?
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- $e^{-x}$ approaches $e^0 = 1$ as $x \to 0$.
- $e^{-x}$ approaches $e^0 = 1$ as $x \to 0$.
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What is indicated by $\lim_{x \to c} f(x) \neq f(c)$?
What is indicated by $\lim_{x \to c} f(x) \neq f(c)$?
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A removable discontinuity at $x = c$. The limit exists but doesn't equal the function value.
A removable discontinuity at $x = c$. The limit exists but doesn't equal the function value.
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What does $\lim_{x \to a} f(x)$ equal if $f(x)$ is linear?
What does $\lim_{x \to a} f(x)$ equal if $f(x)$ is linear?
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$f(a)$. Linear functions are continuous everywhere.
$f(a)$. Linear functions are continuous everywhere.
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Identify $\lim_{x \to 0^-} \frac{1}{x}$.
Identify $\lim_{x \to 0^-} \frac{1}{x}$.
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$-\infty$. Approaching from the left gives negative infinity.
$-\infty$. Approaching from the left gives negative infinity.
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Identify $\lim_{x \to -\infty} f(x)$ from the graph.
Identify $\lim_{x \to -\infty} f(x)$ from the graph.
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The left horizontal asymptote value if it exists. Check where the graph levels off on the left side.
The left horizontal asymptote value if it exists. Check where the graph levels off on the left side.
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What does $\lim_{x \to a} f(x) = f(a)$ imply?
What does $\lim_{x \to a} f(x) = f(a)$ imply?
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Function is continuous at $x = a$. The limit equals the function value at that point.
Function is continuous at $x = a$. The limit equals the function value at that point.
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How do you find $\lim_{x \to 5} f(x)$ if $f(x)$ is constant?
How do you find $\lim_{x \to 5} f(x)$ if $f(x)$ is constant?
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The constant value. Constant functions have the same value everywhere.
The constant value. Constant functions have the same value everywhere.
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How do you denote the limit of $f(x)$ as $x$ approaches $a$ from the right?
How do you denote the limit of $f(x)$ as $x$ approaches $a$ from the right?
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$\lim_{x \to a^+} f(x)$. The plus superscript denotes right-hand limit notation.
$\lim_{x \to a^+} f(x)$. The plus superscript denotes right-hand limit notation.
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What is the behavior of $f(x)$ near $x = a$ if $\lim_{x \to a} f(x) = L$?
What is the behavior of $f(x)$ near $x = a$ if $\lim_{x \to a} f(x) = L$?
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Approaches $L$ as $x$ approaches $a$. This describes the fundamental limit behavior.
Approaches $L$ as $x$ approaches $a$. This describes the fundamental limit behavior.
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What is $\lim_{x \to 0} \sin(\frac{1}{x})$?
What is $\lim_{x \to 0} \sin(\frac{1}{x})$?
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The limit does not exist due to oscillation. The function oscillates infinitely near $x = 0$.
The limit does not exist due to oscillation. The function oscillates infinitely near $x = 0$.
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What does a continuous graph imply about limits at all points?
What does a continuous graph imply about limits at all points?
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Limits exist and equal the function values. Continuity guarantees limit existence everywhere.
Limits exist and equal the function values. Continuity guarantees limit existence everywhere.
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Determine $\lim_{x \to \infty} \frac{1}{x^2}$ using a graph.
Determine $\lim_{x \to \infty} \frac{1}{x^2}$ using a graph.
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- $\frac{1}{x^2}$ approaches $0$ as $x$ becomes large.
- $\frac{1}{x^2}$ approaches $0$ as $x$ becomes large.
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Which term describes the behavior of $f(x)$ as $x \to 2^+$?
Which term describes the behavior of $f(x)$ as $x \to 2^+$?
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Right-hand limit. The $+$ superscript indicates approach from the right side.
Right-hand limit. The $+$ superscript indicates approach from the right side.
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Determine $\lim_{x \to 0} x^2$ using a graph.
Determine $\lim_{x \to 0} x^2$ using a graph.
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- The function $x^2$ approaches $0$ as $x$ approaches $0$.
- The function $x^2$ approaches $0$ as $x$ approaches $0$.
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What is the result of $\lim_{x \to 2^-} f(x)$ if $f(x)$ jumps at $x = 2$?
What is the result of $\lim_{x \to 2^-} f(x)$ if $f(x)$ jumps at $x = 2$?
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The $y$-value approaching from the left of $x = 2$. One-sided limits can exist even with jump discontinuities.
The $y$-value approaching from the left of $x = 2$. One-sided limits can exist even with jump discontinuities.
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Determine $\lim_{x \to 4} f(x)$ if $f(x)$ is continuous at $x = 4$.
Determine $\lim_{x \to 4} f(x)$ if $f(x)$ is continuous at $x = 4$.
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$f(4)$. Continuity means the limit equals the function value.
$f(4)$. Continuity means the limit equals the function value.
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How does $\lim_{x \to 0} \frac{1}{x^2}$ behave?
How does $\lim_{x \to 0} \frac{1}{x^2}$ behave?
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Approaches $\infty$. $\frac{1}{x^2}$ grows without bound near $x = 0$.
Approaches $\infty$. $\frac{1}{x^2}$ grows without bound near $x = 0$.
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What is $\lim_{x \to c} f(x)$ if $f(x)$ is undefined at $x = c$?
What is $\lim_{x \to c} f(x)$ if $f(x)$ is undefined at $x = c$?
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The $y$-value $f(x)$ approaches as $x$ nears $c$. Limits exist even when function values don't.
The $y$-value $f(x)$ approaches as $x$ nears $c$. Limits exist even when function values don't.
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State the notation for the left-hand limit of $f(x)$ at $x = a$.
State the notation for the left-hand limit of $f(x)$ at $x = a$.
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$\lim_{x \to a^-} f(x)$. Standard notation for left-hand limits.
$\lim_{x \to a^-} f(x)$. Standard notation for left-hand limits.
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State the meaning of $\lim_{x \to c} f(x) = L$.
State the meaning of $\lim_{x \to c} f(x) = L$.
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$f(x)$ approaches $L$ as $x$ approaches $c$. This is the formal definition of a limit.
$f(x)$ approaches $L$ as $x$ approaches $c$. This is the formal definition of a limit.
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How do you denote the limit of $f(x)$ as $x$ approaches $a$ on a graph?
How do you denote the limit of $f(x)$ as $x$ approaches $a$ on a graph?
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$\lim_{x \to a} f(x)$. Standard mathematical notation for limits.
$\lim_{x \to a} f(x)$. Standard mathematical notation for limits.
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What does $\lim_{x \to c^-} f(x)$ represent?
What does $\lim_{x \to c^-} f(x)$ represent?
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Limit from the left of $c$. The minus superscript indicates approach from the left.
Limit from the left of $c$. The minus superscript indicates approach from the left.
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What does $\lim_{x \to \infty} \frac{1}{x}$ equal?
What does $\lim_{x \to \infty} \frac{1}{x}$ equal?
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- $\frac{1}{x}$ approaches $0$ as $x$ becomes large.
- $\frac{1}{x}$ approaches $0$ as $x$ becomes large.
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How do you find $\lim_{x \to \infty} f(x)$ from a graph?
How do you find $\lim_{x \to \infty} f(x)$ from a graph?
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Identify the horizontal asymptote if it exists. Look for the horizontal line the graph approaches.
Identify the horizontal asymptote if it exists. Look for the horizontal line the graph approaches.
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What is the limit as $x \to \infty$ of a linear function?
What is the limit as $x \to \infty$ of a linear function?
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Does not exist; increases or decreases without bound. Linear functions grow without bound at infinity.
Does not exist; increases or decreases without bound. Linear functions grow without bound at infinity.
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State the vertical asymptote implication for limits.
State the vertical asymptote implication for limits.
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Limit does not exist at the vertical asymptote. Vertical asymptotes prevent limit existence.
Limit does not exist at the vertical asymptote. Vertical asymptotes prevent limit existence.
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