Math / Calculus

QuestionThe graph below is the function $f(x)$
Find $\lim_{x \to 4^-} f(x)$
Find $\lim_{x \to 4^+} f(x)$
Find $\lim_{x \to 4} f(x)$ Enter an integer or decimal number [more..]

Studdy Solution

STEP 1

What is this asking? We need to find the left-hand limit, right-hand limit, and the limit of a function as $x$ approaches 4, using its graph. Watch out! Don't mix up left and right limits, and remember a limit only exists if both sides agree!

STEP 2

1. Left-hand Limit
2. Right-hand Limit
3. Limit

STEP 3

Let's carefully look at the graph as $x$ approaches 4 from the left side (values smaller than 4).
Imagine a little bug crawling along the graph towards $x = \textbf{4}$ from the left.

STEP 4

As our bug gets closer and closer to $x = \textbf{4}$ from the left, the function's value seems to be oscillating wildly between 2 and 4.
This means it doesn't settle on a single value!

STEP 5

Since the function doesn't approach a single value as $x$ approaches 4 from the left, the left-hand limit doesn't exist.
We write this as: $\lim_{x \to 4^-} f(x) \text{ does not exist}$

STEP 6

Now, let's see what happens as $x$ approaches 4 from the right side (values bigger than 4).
Imagine our bug now crawling from the right towards $x = \textbf{4}$ .

STEP 7

As $x$ gets closer and closer to 4 from the right, the function's value approaches 3.
It's like our bug is heading straight for a y-value of 3!

STEP 8

So, the right-hand limit is 3.
Mathematically, we write: $\lim_{x \to 4^+} f(x) = 3$

STEP 9

For the limit to exist at $x = \textbf{4}$ , both the left-hand and right-hand limits must exist and be equal.

STEP 10

We found that the left-hand limit doesn't exist, and the right-hand limit is 3.
Since they don't match (one doesn't even exist!), the overall limit at $x = \textbf{4}$ doesn't exist!

STEP 11

We write this as: $\lim_{x \to 4} f(x) \text{ does not exist}$

STEP 12

$\lim_{x \to 4^-} f(x) \text{ does not exist}$ $\lim_{x \to 4^+} f(x) = 3$ $\lim_{x \to 4} f(x) \text{ does not exist}$

Was this helpful?

QuestionThe graph below is the function $f(x)$
Find $\lim_{x \to 4^-} f(x)$
Find $\lim_{x \to 4^+} f(x)$
Find $\lim_{x \to 4} f(x)$ Enter an integer or decimal number [more..]

Studdy Solution

STEP 1

What is this asking? We need to find the left-hand limit, right-hand limit, and the limit of a function as $x$ approaches 4, using its graph. Watch out! Don't mix up left and right limits, and remember a limit only exists if both sides agree!

STEP 2

1. Left-hand Limit
2. Right-hand Limit
3. Limit

STEP 3

Let's carefully look at the graph as $x$ approaches 4 from the left side (values smaller than 4).
Imagine a little bug crawling along the graph towards $x = \textbf{4}$ from the left.

STEP 4

As our bug gets closer and closer to $x = \textbf{4}$ from the left, the function's value seems to be oscillating wildly between 2 and 4.
This means it doesn't settle on a single value!

STEP 5

Since the function doesn't approach a single value as $x$ approaches 4 from the left, the left-hand limit doesn't exist.
We write this as: $\lim_{x \to 4^-} f(x) \text{ does not exist}$

STEP 6

Now, let's see what happens as $x$ approaches 4 from the right side (values bigger than 4).
Imagine our bug now crawling from the right towards $x = \textbf{4}$ .

STEP 7

As $x$ gets closer and closer to 4 from the right, the function's value approaches 3.
It's like our bug is heading straight for a y-value of 3!

STEP 8

So, the right-hand limit is 3.
Mathematically, we write: $\lim_{x \to 4^+} f(x) = 3$

STEP 9

For the limit to exist at $x = \textbf{4}$ , both the left-hand and right-hand limits must exist and be equal.

STEP 10

We found that the left-hand limit doesn't exist, and the right-hand limit is 3.
Since they don't match (one doesn't even exist!), the overall limit at $x = \textbf{4}$ doesn't exist!

STEP 11

We write this as: $\lim_{x \to 4} f(x) \text{ does not exist}$

STEP 12

$\lim_{x \to 4^-} f(x) \text{ does not exist}$ $\lim_{x \to 4^+} f(x) = 3$ $\lim_{x \to 4} f(x) \text{ does not exist}$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionThe graph below is the function f(x)f(x)f(x)Find lim⁡x→4−f(x)\lim_{x \to 4^-} f(x)limx→4−​f(x)Find lim⁡x→4+f(x)\lim_{x \to 4^+} f(x)limx→4+​f(x)Find lim⁡x→4f(x)\lim_{x \to 4} f(x)limx→4​f(x) Enter an integer or decimal number [more..]

Studdy Solution

STEP 1

What is this asking? We need to find the left-hand limit, right-hand limit, and the limit of a function as xxx approaches **4**, using its graph. Watch out! Don't mix up left and right limits, and remember a limit only exists if both sides agree!

STEP 2

1. Left-hand Limit2. Right-hand Limit3. Limit

STEP 3

Let's **carefully** look at the graph as xxx approaches **4** from the **left** side (values smaller than **4**).Imagine a little bug crawling along the graph towards x=4x = \textbf{4}x=4 from the left.

STEP 4

As our bug gets closer and closer to x=4x = \textbf{4}x=4 from the left, the function's value seems to be oscillating wildly between **2** and **4**.This means it doesn't settle on a single value!

STEP 5

Since the function doesn't approach a single value as xxx approaches **4** from the left, the left-hand limit *doesn't exist*.We write this as: lim⁡x→4−f(x) does not exist \lim_{x \to 4^-} f(x) \text{ does not exist} x→4−lim​f(x) does not exist

STEP 6

Now, let's see what happens as xxx approaches **4** from the **right** side (values bigger than **4**).Imagine our bug now crawling from the right towards x=4x = \textbf{4}x=4.

STEP 7

As xxx gets closer and closer to **4** from the right, the function's value approaches **3**.It's like our bug is heading straight for a y-value of **3**!

STEP 8

So, the right-hand limit is **3**.Mathematically, we write: lim⁡x→4+f(x)=3 \lim_{x \to 4^+} f(x) = 3 x→4+lim​f(x)=3

STEP 9

For the limit to exist at x=4x = \textbf{4}x=4, *both* the left-hand and right-hand limits must exist and be equal.

STEP 10

We found that the left-hand limit *doesn't exist*, and the right-hand limit is **3**.Since they don't match (one doesn't even exist!), the overall limit at x=4x = \textbf{4}x=4 *doesn't exist*!

STEP 11

We write this as: lim⁡x→4f(x) does not exist \lim_{x \to 4} f(x) \text{ does not exist} x→4lim​f(x) does not exist

STEP 12

lim⁡x→4−f(x) does not exist \lim_{x \to 4^-} f(x) \text{ does not exist} x→4−lim​f(x) does not exist lim⁡x→4+f(x)=3 \lim_{x \to 4^+} f(x) = 3 x→4+lim​f(x)=3 lim⁡x→4f(x) does not exist \lim_{x \to 4} f(x) \text{ does not exist} x→4lim​f(x) does not exist

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionThe graph below is the function $f(x)$
Find $\lim_{x \to 4^-} f(x)$
Find $\lim_{x \to 4^+} f(x)$
Find $\lim_{x \to 4} f(x)$ Enter an integer or decimal number [more..]

What is this asking? We need to find the left-hand limit, right-hand limit, and the limit of a function as $x$ approaches 4, using its graph. Watch out! Don't mix up left and right limits, and remember a limit only exists if both sides agree!

1. Left-hand Limit
2. Right-hand Limit
3. Limit

Let's carefully look at the graph as $x$ approaches 4 from the left side (values smaller than 4).
Imagine a little bug crawling along the graph towards $x = \textbf{4}$ from the left.

As our bug gets closer and closer to $x = \textbf{4}$ from the left, the function's value seems to be oscillating wildly between 2 and 4.
This means it doesn't settle on a single value!

Since the function doesn't approach a single value as $x$ approaches 4 from the left, the left-hand limit doesn't exist.
We write this as: $\lim_{x \to 4^-} f(x) \text{ does not exist}$

Now, let's see what happens as $x$ approaches 4 from the right side (values bigger than 4).
Imagine our bug now crawling from the right towards $x = \textbf{4}$ .

As $x$ gets closer and closer to 4 from the right, the function's value approaches 3.
It's like our bug is heading straight for a y-value of 3!

So, the right-hand limit is 3.
Mathematically, we write: $\lim_{x \to 4^+} f(x) = 3$

For the limit to exist at $x = \textbf{4}$ , both the left-hand and right-hand limits must exist and be equal.

We found that the left-hand limit doesn't exist, and the right-hand limit is 3.
Since they don't match (one doesn't even exist!), the overall limit at $x = \textbf{4}$ doesn't exist!

We write this as: $\lim_{x \to 4} f(x) \text{ does not exist}$

$\lim_{x \to 4^-} f(x) \text{ does not exist}$ $\lim_{x \to 4^+} f(x) = 3$ $\lim_{x \to 4} f(x) \text{ does not exist}$