QuestionGiven the table values of $f(x)$ for $x$ near 4, which limit conclusion is correct? (A) $\lim _{x \rightarrow 4} f(x)=6$ (B) $\lim _{x \rightarrow 4} f(x)=7$ (C) $\lim _{x \rightarrow 4^{-}} f(x)=6$ and $\lim _{x \rightarrow 4^{+}} f(x)=7$ (D) $\lim _{x \rightarrow 4^{-}} f(x)=7$ and $\lim _{x \rightarrow 4^{+}} f(x)=6$

Studdy Solution

STEP 1

Assumptions1. The table provides values of function $f$ at selected values of $x$ . . We are interested in the limit of the function $f$ as $x$ approaches4 from both sides.

STEP 2

First, we need to understand the concept of the limit of a function at a point. The limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value.

STEP 3

Now, let's look at the limit of the function $f$ as $x$ approaches from the left side (denoted as $x \rightarrow^{-}$ ). This is the value that $f(x)$ is approaching as $x$ gets closer and closer to from values less than.

STEP 4

From the table, we can see that as $x$ gets closer and closer to4 from the left (values less than4), $f(x)$ is getting closer and closer to6.

STEP 5

Therefore, we can conclude that $\lim{x \rightarrow4^{-}} f(x)=$

STEP 6

Next, let's look at the limit of the function $f$ as $x$ approaches4 from the right side (denoted as $x \rightarrow4^{+}$ ). This is the value that $f(x)$ is approaching as $x$ gets closer and closer to4 from values greater than4.

STEP 7

From the table, we can see that as $x$ gets closer and closer to4 from the right (values greater than4), $f(x)$ is getting closer and closer to7.

STEP 8

Therefore, we can conclude that $\lim{x \rightarrow4^{+}} f(x)=7$

STEP 9

Since the limit of the function $f$ as $x$ approaches4 from the left is not equal to the limit of the function $f$ as $x$ approaches4 from the right, we can say that the limit of the function $f$ at $x=4$ does not exist.

STEP 10

Therefore, the conclusion supported by the data in the table is that $\lim{x \rightarrow4^{-}} f(x)=6$ and $\lim{x \rightarrow4^{+}} f(x)=7$ , which corresponds to option (C).
The answer is (C) $\lim{x \rightarrow4^{-}} f(x)=6$ and $\lim{x \rightarrow4^{+}} f(x)=7$ .

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Studdy Solution

STEP 1

Assumptions1. The table provides values of function $f$ at selected values of $x$ . . We are interested in the limit of the function $f$ as $x$ approaches4 from both sides.

STEP 2

First, we need to understand the concept of the limit of a function at a point. The limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value.

STEP 3

Now, let's look at the limit of the function $f$ as $x$ approaches from the left side (denoted as $x \rightarrow^{-}$ ). This is the value that $f(x)$ is approaching as $x$ gets closer and closer to from values less than.

STEP 4

From the table, we can see that as $x$ gets closer and closer to4 from the left (values less than4), $f(x)$ is getting closer and closer to6.

STEP 5

Therefore, we can conclude that $\lim{x \rightarrow4^{-}} f(x)=$

STEP 6

Next, let's look at the limit of the function $f$ as $x$ approaches4 from the right side (denoted as $x \rightarrow4^{+}$ ). This is the value that $f(x)$ is approaching as $x$ gets closer and closer to4 from values greater than4.

STEP 7

From the table, we can see that as $x$ gets closer and closer to4 from the right (values greater than4), $f(x)$ is getting closer and closer to7.

STEP 8

Therefore, we can conclude that $\lim{x \rightarrow4^{+}} f(x)=7$

STEP 9

Since the limit of the function $f$ as $x$ approaches4 from the left is not equal to the limit of the function $f$ as $x$ approaches4 from the right, we can say that the limit of the function $f$ at $x=4$ does not exist.

STEP 10

Therefore, the conclusion supported by the data in the table is that $\lim{x \rightarrow4^{-}} f(x)=6$ and $\lim{x \rightarrow4^{+}} f(x)=7$ , which corresponds to option (C).
The answer is (C) $\lim{x \rightarrow4^{-}} f(x)=6$ and $\lim{x \rightarrow4^{+}} f(x)=7$ .

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Studdy Solution

STEP 1

Assumptions1. The table provides values of function fff at selected values of xxx. . We are interested in the limit of the function fff as xxx approaches4 from both sides.

STEP 2

First, we need to understand the concept of the limit of a function at a point. The limit of a function as xxx approaches a certain value is the value that the function approaches as xxx gets closer and closer to that value.

STEP 3

Now, let's look at the limit of the function fff as xxx approaches from the left side (denoted as x→−x \rightarrow^{-}x→−). This is the value that f(x)f(x)f(x) is approaching as xxx gets closer and closer to from values less than.

STEP 4

From the table, we can see that as xxx gets closer and closer to4 from the left (values less than4), f(x)f(x)f(x) is getting closer and closer to6.

STEP 5

Therefore, we can conclude that lim⁡x→4−f(x)=\lim{x \rightarrow4^{-}} f(x)=limx→4−f(x)=

STEP 6

Next, let's look at the limit of the function fff as xxx approaches4 from the right side (denoted as x→4+x \rightarrow4^{+}x→4+). This is the value that f(x)f(x)f(x) is approaching as xxx gets closer and closer to4 from values greater than4.

STEP 7

From the table, we can see that as xxx gets closer and closer to4 from the right (values greater than4), f(x)f(x)f(x) is getting closer and closer to7.

STEP 8

Therefore, we can conclude that lim⁡x→4+f(x)=7\lim{x \rightarrow4^{+}} f(x)=7limx→4+f(x)=7

STEP 9

Since the limit of the function fff as xxx approaches4 from the left is not equal to the limit of the function fff as xxx approaches4 from the right, we can say that the limit of the function fff at x=4x=4x=4 does not exist.

STEP 10

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Assumptions1. The table provides values of function $f$ at selected values of $x$ . . We are interested in the limit of the function $f$ as $x$ approaches4 from both sides.

First, we need to understand the concept of the limit of a function at a point. The limit of a function as $x$ approaches a certain value is the value that the function approaches as $x$ gets closer and closer to that value.

Now, let's look at the limit of the function $f$ as $x$ approaches from the left side (denoted as $x \rightarrow^{-}$ ). This is the value that $f(x)$ is approaching as $x$ gets closer and closer to from values less than.

From the table, we can see that as $x$ gets closer and closer to4 from the left (values less than4), $f(x)$ is getting closer and closer to6.

Therefore, we can conclude that $\lim{x \rightarrow4^{-}} f(x)=$

Next, let's look at the limit of the function $f$ as $x$ approaches4 from the right side (denoted as $x \rightarrow4^{+}$ ). This is the value that $f(x)$ is approaching as $x$ gets closer and closer to4 from values greater than4.

From the table, we can see that as $x$ gets closer and closer to4 from the right (values greater than4), $f(x)$ is getting closer and closer to7.

Therefore, we can conclude that $\lim{x \rightarrow4^{+}} f(x)=7$

Since the limit of the function $f$ as $x$ approaches4 from the left is not equal to the limit of the function $f$ as $x$ approaches4 from the right, we can say that the limit of the function $f$ at $x=4$ does not exist.