Podcast
Questions and Answers
When evaluating $\lim_{x \to a} f(x)$, what is the most important factor in determining if the limit exists?
When evaluating $\lim_{x \to a} f(x)$, what is the most important factor in determining if the limit exists?
- The function $f(x)$ must be continuous at $x = a$.
- The value of $f(a)$ must be defined.
- The function $f(x)$ must be differentiable at $x = a$.
- The left-hand limit and the right-hand limit at $x = a$ must both exist and be equal. (correct)
Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. What is $\lim_{x \to 2} f(x)$?
Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. What is $\lim_{x \to 2} f(x)$?
- The limit does not exist because the function is undefined at $x = 2$.
- 4 (correct)
- 2
- 0
For a function to be continuous at a point $x = c$, which of the following conditions must be met?
For a function to be continuous at a point $x = c$, which of the following conditions must be met?
- The derivative of $f(x)$ exists at $x = c$.
- $\lim_{x \to c} f(x)$ exists and is finite
- $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$. (correct)
- $f(c)$ is defined.
Given the function $f(x) = \begin{cases} x^2, & x < 1 \ 3x - 2, & x \ge 1 \end{cases}$, is $f(x)$ continuous at $x = 1$?
Given the function $f(x) = \begin{cases} x^2, & x < 1 \ 3x - 2, & x \ge 1 \end{cases}$, is $f(x)$ continuous at $x = 1$?
Determine the interval(s) on which the function $f(x) = \sqrt{9 - x^2}$ is continuous.
Determine the interval(s) on which the function $f(x) = \sqrt{9 - x^2}$ is continuous.
Flashcards
Limit of a function
Limit of a function
The value that a function approaches as the input approaches a certain value.
Continuity of a function
Continuity of a function
A function is continuous if its graph can be drawn without lifting the pen.
Interval
Interval
A range of real numbers between two points.
Conditions of continuity
Conditions of continuity
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Value of limits
Value of limits
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Study Notes
Limits and Continuity
- Limits: A limit describes the value a function approaches as the input (often x) gets closer to a particular value.
- Finding Limits: Various methods can be used to determine limits, but the examples provided demonstrate evaluating expressions as the input approaches a specific value, often using substitution.
- Example 1: The limit of (x + 2) as x approaches 3 would be 5 (substitution).
- Example 2: The limit of (3x² + 4x - 7) as x approaches -2 would be -1 (substitution).
- Example 3: The limit of (x + 1)(x - 3) as x approaches 2 is -3 (substitution).
- Continuity: A function is continuous if there's no break or gap in its graph. The graph can be drawn without lifting the pen.
Conditions for Continuity
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Existence: The function must exist at the given value.
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Limit Exists at Given Point: The limit of the function as x approaches the value must be defined.
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Function Value Equal to Limit: The value of the function at the point must equal the limit.
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Example: If the function satisfies these 3 criteria at a certain value x, then the function would be considered continuous.
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Description
Explore limits, which describe a function's approached value as the input nears a specific value. Learn about continuity, where a function has no breaks and can be drawn without lifting the pen. Understand the conditions required for continuity.
