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How does a hole appear on a graph?
As an open circle at a specific point, indicating the function is not defined there.
What does the graph of a rational function with a hole tell you about its limit at that point?
The y-value the function approaches as x approaches the hole's x-coordinate is the limit of the function at that point.
How can you visually differentiate between a hole and a vertical asymptote on a graph?
A hole is a single point discontinuity (open circle), while a vertical asymptote is a line the function approaches but never crosses, causing the function to tend towards infinity.
What does the smoothness of the graph around a hole indicate?
It indicates that the limit exists at that point, even though the function is undefined.
How to identify a hole from a graph?
Look for an open circle (or removable discontinuity) on the graph.
What does a hole on a graph imply about the function's domain?
The function's domain excludes the x-value of the hole.
What does it mean if a graph has a hole at x=a?
The function is not defined at x=a, but the limit as x approaches 'a' exists.
How does the graph of a simplified rational function relate to the original function with a hole?
The graph of the simplified function is identical to the original function, except it is defined at the x-value where the hole existed.
What does the presence of a hole suggest about the function's continuity?
The function is discontinuous at the x-value of the hole, but it's a removable discontinuity.
How to estimate the limit at a hole using a graph?
Trace the graph towards the x-value of the hole from both sides; the y-value the graph approaches is the estimated limit.
How to find holes in a rational function?
1. Factor numerator and denominator. 2. Identify common factors. 3. Set common factor = 0 to find x-coordinate. 4. Cancel common factors. 5. Evaluate simplified function at x-coordinate to find y-coordinate.
Steps to find the limit at a hole.
1. Identify the x-coordinate of the hole. 2. Cancel the common factor. 3. Evaluate the simplified function at the x-coordinate.
How to determine if a rational function has a hole at x=a?
1. Check if (x-a) is a factor of both numerator and denominator. 2. If yes, then a hole exists at x=a.
How to simplify a rational function with a hole?
1. Factor the numerator and the denominator. 2. Identify and cancel the common factors. 3. Write the simplified function.
How to verify the y-coordinate of a hole graphically?
1. Graph the rational function. 2. Zoom in around the x-coordinate of the hole. 3. Observe the y-value the function approaches.
How to deal with multiple holes in a rational function?
1. Factor completely. 2. Identify all common factors. 3. Find coordinates for each hole separately.
What is the first step when analyzing rational functions for holes?
Factor both the numerator and the denominator completely.
How do you find the x-value where a hole occurs?
Set the common factor (that appears in both numerator and denominator) equal to zero and solve for x.
What do you do after canceling the common factor to find the y-coordinate of the hole?
Substitute the x-value of the hole into the simplified rational function.
How to confirm a hole exists at a specific x-value?
Verify that the function is undefined at that x-value due to a common factor, but the limit exists as x approaches that value.
Formula for finding the y-coordinate of a hole at x = a.
Given r(x) = p(x)/q(x) with a hole at x=a, y = lim x->a [r(x) after canceling common factors].
Express the limit at a hole mathematically.
If r(x) has a hole at x = c, then $\lim_{x \to c} r(x) = L$, where L is the y-coordinate of the hole.
Factoring formula for $x^2 - a^2$.
$x^2 - a^2 = (x - a)(x + a)$
General form of a rational function with a hole at x=a.
$\frac{(x-a)p(x)}{(x-a)q(x)}$, where p(a) and q(a) are not simultaneously zero.
How to find the limit L at a hole x=c?
L = $\lim_{x \to c}$ [Simplified r(x) after canceling common factors]
Write the general form of a factor in a polynomial.
$(x - a)$, where 'a' is a root of the polynomial.
How to express the y-coordinate of a hole using the simplified function?
If the simplified function is s(x), then y-coordinate = s(a), where x=a is the x-coordinate of the hole.
Formula for factoring a cubic difference $x^3 - a^3$.
$x^3 - a^3 = (x - a)(x^2 + ax + a^2)$
Formula for factoring a cubic sum $x^3 + a^3$.
$x^3 + a^3 = (x + a)(x^2 - ax + a^2)$
Express the condition for a hole using limits.
$\lim_{x \to a} f(x)$ exists, but $f(a)$ is undefined.