Math / Calculus

Question5. $\lim _{x \rightarrow 2^{-}} \frac{x}{2-x}$

Studdy Solution

STEP 1

What is this asking? What happens to the fraction $\frac{x}{2-x}$ as $x$ gets really close to $2$ from the left? Watch out! We're approaching $2$ from the left, meaning $x$ is slightly less than $2$ , not greater!

STEP 2

1. Analyze the numerator
2. Analyze the denominator
3. Combine the analysis

STEP 3

As $x$ approaches $2$ from the left, the numerator, $x$ , gets closer and closer to $\textbf{2}$ .
It's like walking towards a finish line – you're getting closer and closer to your destination!

STEP 4

Since we're approaching $2$ from the left, $x$ is a little less than $2$ .
Think of it as $x = 2 - \text{tiny bit}$ .

STEP 5

So, the denominator, $2 - x$ , becomes $2 - (2 - \text{tiny bit})$ .

STEP 6

This simplifies to $\textbf{tiny bit}$ .
The denominator is a small positive number getting closer and closer to zero!

STEP 7

We have a numerator approaching $\textbf{2}$ and a denominator, a tiny positive number, approaching $\textbf{0}$ .
Imagine dividing $2$ by a super tiny number, like $0.000001$ .
The result is a huge number!

STEP 8

As the "tiny bit" gets even tinier, the fraction $\frac{2}{\text{tiny bit}}$ gets larger and larger.
It grows without bound!

STEP 9

Therefore, the limit is positive infinity.

STEP 10

$\lim_{x \rightarrow 2^{-}} \frac{x}{2-x} = \infty$

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Question5. $\lim _{x \rightarrow 2^{-}} \frac{x}{2-x}$

Studdy Solution

STEP 1

What is this asking? What happens to the fraction $\frac{x}{2-x}$ as $x$ gets really close to $2$ from the left? Watch out! We're approaching $2$ from the left, meaning $x$ is slightly less than $2$ , not greater!

STEP 2

1. Analyze the numerator
2. Analyze the denominator
3. Combine the analysis

STEP 3

As $x$ approaches $2$ from the left, the numerator, $x$ , gets closer and closer to $\textbf{2}$ .
It's like walking towards a finish line – you're getting closer and closer to your destination!

STEP 4

Since we're approaching $2$ from the left, $x$ is a little less than $2$ .
Think of it as $x = 2 - \text{tiny bit}$ .

STEP 5

So, the denominator, $2 - x$ , becomes $2 - (2 - \text{tiny bit})$ .

STEP 6

This simplifies to $\textbf{tiny bit}$ .
The denominator is a small positive number getting closer and closer to zero!

STEP 7

We have a numerator approaching $\textbf{2}$ and a denominator, a tiny positive number, approaching $\textbf{0}$ .
Imagine dividing $2$ by a super tiny number, like $0.000001$ .
The result is a huge number!

STEP 8

As the "tiny bit" gets even tinier, the fraction $\frac{2}{\text{tiny bit}}$ gets larger and larger.
It grows without bound!

STEP 9

Therefore, the limit is positive infinity.

STEP 10

$\lim_{x \rightarrow 2^{-}} \frac{x}{2-x} = \infty$

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Question5. lim⁡x→2−x2−x\lim _{x \rightarrow 2^{-}} \frac{x}{2-x}limx→2−​2−xx​

Studdy Solution

STEP 1

What is this asking? What happens to the fraction x2−x \frac{x}{2-x} 2−xx​ as xxx gets really close to 222 from the *left*? Watch out! We're approaching 222 from the *left*, meaning xxx is slightly less than 222, not greater!

STEP 2

1. Analyze the numerator2. Analyze the denominator3. Combine the analysis

STEP 3

As xxx approaches 222 from the left, the numerator, xxx, gets closer and closer to 2\textbf{2}2.It's like walking towards a finish line – you're getting closer and closer to your destination!

STEP 4

Since we're approaching 222 from the *left*, xxx is a little less than 222.Think of it as x=2−tiny bitx = 2 - \text{tiny bit}x=2−tiny bit.

STEP 5

So, the denominator, 2−x2 - x2−x, becomes 2−(2−tiny bit)2 - (2 - \text{tiny bit})2−(2−tiny bit).

STEP 6

This simplifies to tiny bit\textbf{tiny bit}tiny bit.The denominator is a small positive number getting closer and closer to zero!

STEP 7

We have a numerator approaching 2\textbf{2}2 and a denominator, a tiny positive number, approaching 0\textbf{0}0.Imagine dividing 222 by a super tiny number, like 0.0000010.0000010.000001.The result is a huge number!

STEP 8

As the "tiny bit" gets even tinier, the fraction 2tiny bit\frac{2}{\text{tiny bit}}tiny bit2​ gets larger and larger.It grows without bound!

STEP 9

Therefore, the limit is positive infinity.

STEP 10

lim⁡x→2−x2−x=∞ \lim_{x \rightarrow 2^{-}} \frac{x}{2-x} = \infty x→2−lim​2−xx​=∞

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Question5. $\lim _{x \rightarrow 2^{-}} \frac{x}{2-x}$

What is this asking? What happens to the fraction $\frac{x}{2-x}$ as $x$ gets really close to $2$ from the left? Watch out! We're approaching $2$ from the left, meaning $x$ is slightly less than $2$ , not greater!

1. Analyze the numerator
2. Analyze the denominator
3. Combine the analysis

As $x$ approaches $2$ from the left, the numerator, $x$ , gets closer and closer to $\textbf{2}$ .
It's like walking towards a finish line – you're getting closer and closer to your destination!

Since we're approaching $2$ from the left, $x$ is a little less than $2$ .
Think of it as $x = 2 - \text{tiny bit}$ .

So, the denominator, $2 - x$ , becomes $2 - (2 - \text{tiny bit})$ .

This simplifies to $\textbf{tiny bit}$ .
The denominator is a small positive number getting closer and closer to zero!

We have a numerator approaching $\textbf{2}$ and a denominator, a tiny positive number, approaching $\textbf{0}$ .
Imagine dividing $2$ by a super tiny number, like $0.000001$ .
The result is a huge number!

As the "tiny bit" gets even tinier, the fraction $\frac{2}{\text{tiny bit}}$ gets larger and larger.
It grows without bound!

$\lim_{x \rightarrow 2^{-}} \frac{x}{2-x} = \infty$