Limits Language, Notation, And Features Of Functions

1) Match the following

continuous at

essential discontinuity at

jump discontinuity at

removable discontinuity at

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2) Find the limit (number, ∞, -∞) or write "DNE" (does not exist). ${lim}_{x\to 1}f(x)$ : _____

The limit of the given expression is 0.

Explanation

The limit of the given expression is 0.

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4) Find the limit (number, ∞, -∞) or write "DNE" (does not exist). ${lim}_{x\to \infty }f(x)$ : _____

The answer to the question is -∞, which represents negative infinity. In the context of limits, when the limit of a function approaches negative infinity, it means that the function becomes arbitrarily small and negative as the input values approach negative infinity. This indicates that the function has no lower bound and can decrease indefinitely. Therefore, the limit in this case is negative infinity.

Explanation

The answer to the question is -∞, which represents negative infinity. In the context of limits, when the limit of a function approaches negative infinity, it means that the function becomes arbitrarily small and negative as the input values approach negative infinity. This indicates that the function has no lower bound and can decrease indefinitely. Therefore, the limit in this case is negative infinity.

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5) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to \infty }h(x)=-\infty$

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The limit information described in the question refers to the behavior of a function as the input approaches positive or negative infinity. This is known as the end behavior of the function. It helps determine if the function approaches a particular value or goes to positive or negative infinity as the input becomes very large or very small. Therefore, the correct answer is end behavior.

Explanation

The limit information described in the question refers to the behavior of a function as the input approaches positive or negative infinity. This is known as the end behavior of the function. It helps determine if the function approaches a particular value or goes to positive or negative infinity as the input becomes very large or very small. Therefore, the correct answer is end behavior.

6) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to -\infty }G(x)=-3$

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The given correct answer states that the features described by the limit information are "end behavior" and "horizontal asymptote". This means that the limits of the function as it approaches positive or negative infinity are being described, and there is a horizontal line that the function approaches as x goes to infinity or negative infinity.

Explanation

The given correct answer states that the features described by the limit information are "end behavior" and "horizontal asymptote". This means that the limits of the function as it approaches positive or negative infinity are being described, and there is a horizontal line that the function approaches as x goes to infinity or negative infinity.

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7) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to {3}^{+}}F(x)=-2$ and ${lim}_{x\to {3}^{-}}F(x)=3$

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The correct answer is jump discontinuity. This is because jump discontinuity occurs when there is a sudden jump or gap in the graph of a function at a certain point. It is characterized by two different limit values approaching from the left and right sides of the point, but the function does not approach a single value at that point.

Explanation

The correct answer is jump discontinuity. This is because jump discontinuity occurs when there is a sudden jump or gap in the graph of a function at a certain point. It is characterized by two different limit values approaching from the left and right sides of the point, but the function does not approach a single value at that point.

8) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to \infty }c(x)=4$ and ${lim}_{x\to -\infty }c(x)=3$

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The limit information described in the question refers to the behavior of a function as the input approaches positive or negative infinity. The concept of end behavior is used to describe how the function behaves as x approaches infinity or negative infinity. A horizontal asymptote is a line that the function approaches as x approaches infinity or negative infinity. Therefore, the correct answer is end behavior and horizontal asymptote.

Explanation

The limit information described in the question refers to the behavior of a function as the input approaches positive or negative infinity. The concept of end behavior is used to describe how the function behaves as x approaches infinity or negative infinity. A horizontal asymptote is a line that the function approaches as x approaches infinity or negative infinity. Therefore, the correct answer is end behavior and horizontal asymptote.

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9) Find the limit (number, ∞, -∞) or write "DNE" (does not exist). ${lim}_{x\to {-5}^{+}}f(x)$ : _____

The given question asks for the limit as the number approaches negative infinity. The answer is -∞, which represents negative infinity. This means that as the number gets smaller and smaller, it will approach negative infinity.

Explanation

The given question asks for the limit as the number approaches negative infinity. The answer is -∞, which represents negative infinity. This means that as the number gets smaller and smaller, it will approach negative infinity.

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11) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to 5}q(x)=\infty$

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The given limit information describes both an essential discontinuity and a vertical asymptote. An essential discontinuity occurs when the limit of a function does not exist at a particular point, but the function approaches different values from both sides of that point. In this case, the function has an essential discontinuity. A vertical asymptote, on the other hand, is a vertical line that the graph of a function approaches but never crosses. In this case, the function has a vertical asymptote.

Explanation

The given limit information describes both an essential discontinuity and a vertical asymptote. An essential discontinuity occurs when the limit of a function does not exist at a particular point, but the function approaches different values from both sides of that point. In this case, the function has an essential discontinuity. A vertical asymptote, on the other hand, is a vertical line that the graph of a function approaches but never crosses. In this case, the function has a vertical asymptote.

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12) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to 4}P(x)=11$ but $P(4)=10$

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The limit information described by the given answer, removable discontinuity, refers to a situation where a function has a hole or gap at a certain point but can be filled in to make the function continuous at that point. In other words, the function approaches different values from the left and right sides of the hole, but if the hole is filled in with a single point, the function becomes continuous. This is different from essential discontinuity and jump discontinuity, where the function approaches different values from the left and right sides, but cannot be made continuous by filling in a single point. The other options, end behavior, horizontal asymptote, and vertical asymptote, are not described by the limit information given.

Explanation

The limit information described by the given answer, removable discontinuity, refers to a situation where a function has a hole or gap at a certain point but can be filled in to make the function continuous at that point. In other words, the function approaches different values from the left and right sides of the hole, but if the hole is filled in with a single point, the function becomes continuous. This is different from essential discontinuity and jump discontinuity, where the function approaches different values from the left and right sides, but cannot be made continuous by filling in a single point. The other options, end behavior, horizontal asymptote, and vertical asymptote, are not described by the limit information given.

13) Find the limit (number, ∞, -∞) or write "DNE" (does not exist). ${lim}_{x\to 4}f(x)$ : _____

The given expression is a limit problem with an undefined variable. Since the limit is taken as the variable approaches infinity or negative infinity, the value of the expression is constant and does not change. Therefore, the limit of the expression is -4.

Explanation

The given expression is a limit problem with an undefined variable. Since the limit is taken as the variable approaches infinity or negative infinity, the value of the expression is constant and does not change. Therefore, the limit of the expression is -4.

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14) What feature(s) are described by the limit(s) information below? Check all that apply. ${lim}_{x\to {2}^{+}}g(x)=\infty$ and ${lim}_{x\to {2}^{-}}g(x)$ does not exist

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The given limit information "does not exist" suggests that there is a discontinuity in the function. This type of discontinuity is called an essential discontinuity. Additionally, the fact that there is a vertical asymptote implies that the function approaches infinity or negative infinity as it approaches a certain value. Therefore, the correct answer is essential discontinuity and vertical asymptote.

Explanation

The given limit information "does not exist" suggests that there is a discontinuity in the function. This type of discontinuity is called an essential discontinuity. Additionally, the fact that there is a vertical asymptote implies that the function approaches infinity or negative infinity as it approaches a certain value. Therefore, the correct answer is essential discontinuity and vertical asymptote.

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15) What feature(s) are described by the limit(s) information below? Check all that apply.

and

Essential discontinuity

Jump discontinuity

Removable discontinuity

End behavior

Horizontal asymptote

Vertical asymptote

The given answer is correct because the limit information described in the question includes an "essential discontinuity" and a "vertical asymptote". An essential discontinuity occurs when the limit of a function does not exist at a certain point, and a vertical asymptote is a line that the graph of a function approaches but never crosses. Therefore, both of these features are accurately described by the given limit information.

Explanation

The given answer is correct because the limit information described in the question includes an "essential discontinuity" and a "vertical asymptote". An essential discontinuity occurs when the limit of a function does not exist at a certain point, and a vertical asymptote is a line that the graph of a function approaches but never crosses. Therefore, both of these features are accurately described by the given limit information.

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