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How to identify a removable discontinuity in a rational function?
1. Factor the numerator and denominator. 2. Cancel common factors. 3. The cancelled factor indicates the x-value of the discontinuity. 4. Evaluate the simplified function at that x-value to find the y-value of the 'hole'.
How to find the limit at a removable discontinuity?
1. Factor and simplify the function. 2. Substitute the x-value of the discontinuity into the simplified function.
How to identify a jump discontinuity in a piecewise function?
1. Identify the x-value where the function changes definition. 2. Find the left-hand limit at that x-value. 3. Find the right-hand limit at that x-value. 4. If the limits are different, there's a jump discontinuity.
How to determine if a piecewise function is continuous at a point?
1. Check if the function is defined at the point. 2. Find the left-hand limit. 3. Find the right-hand limit. 4. If the limits exist and are equal to the function's value at that point, it's continuous.
How to identify an asymptote discontinuity?
1. Look for values of x that make the denominator of a rational function equal to zero. 2. Check the limits as x approaches these values from both sides. 3. If the limits approach infinity (or negative infinity), there's an asymptote discontinuity.
How to redefine a function to remove a discontinuity?
1. Identify the discontinuity. 2. Find the limit at that point. 3. Define a new piecewise function that equals the original function everywhere except at the discontinuity, where it equals the limit.
How do you find the value that makes a piecewise function continuous?
1. Set the pieces of the function equal to each other at the point where they meet. 2. Solve for the unknown variable.
How to determine if a function is differentiable at a point of discontinuity?
A function cannot be differentiable at a point of discontinuity.
How to analyze the continuity of a function with absolute values?
1. Rewrite the absolute value function as a piecewise function. 2. Analyze the continuity of the resulting piecewise function.
How to find the equation of a vertical asymptote?
1. Set the denominator of the rational function equal to zero. 2. Solve for x. 3. The solution(s) are the equations of the vertical asymptotes.
Explain the concept of removable discontinuities.
A 'hole' in the graph that can often be 'removed' by simplifying the function (e.g., factoring). The limit exists, but the function is not defined at that point, or defined to a different value.
Explain the concept of jump discontinuities.
A sudden 'jump' in the graph where the left and right-hand limits are different. Common in piecewise functions.
Explain the concept of asymptote discontinuities.
The function approaches infinity (or negative infinity) as x approaches a certain value, creating a vertical asymptote.
Why are removable discontinuities important?
They often involve algebraic manipulation (factoring) to identify and remove them, a skill frequently tested.
Where do jump discontinuities typically occur?
Almost exclusively in piecewise functions, where the different pieces of the function do not connect.
What is the key characteristic of an asymptote discontinuity?
The function's values increase or decrease without bound as x approaches a specific value.
How do you identify a discontinuity in a graph?
Look for breaks, holes, jumps, or vertical asymptotes in the graph.
What is the relationship between limits and continuity?
For a function to be continuous at a point, the limit must exist at that point and be equal to the function's value.
Why is understanding discontinuities important in calculus?
Discontinuities affect the behavior of functions and are crucial for understanding limits, derivatives, and integrals.
How can factoring help identify discontinuities?
Factoring can reveal common factors that lead to removable discontinuities in rational functions.
What is a discontinuity?
A point where a function is not continuous; where you must lift your pencil to draw the graph.
Define a removable discontinuity.
A point where a function is not defined, but the limit exists. Often called a 'hole'.
Define a jump discontinuity.
A point where the left-hand limit does not equal the right-hand limit.
Define an asymptote discontinuity.
A point where the function approaches infinity (or negative infinity).
What is a piecewise function?
A function defined by multiple sub-functions, each applying to a certain interval of the main function's domain.
What is a rational function?
A function that can be expressed as the quotient of two polynomials.
What does it mean for a function to be continuous at a point?
The function is defined at that point, the limit exists at that point, and the limit equals the function's value at that point.
What is a left-hand limit?
The value a function approaches as the input approaches a given value from the left side.
What is a right-hand limit?
The value a function approaches as the input approaches a given value from the right side.
What is the domain of a function?
The set of all possible input values (x-values) for which the function is defined.