Math / Calculus

QuestionQuestion 5
What is the value of $\lim _{x \rightarrow 0^{+}} \frac{1}{2+e^{-\frac{1}{x}}}$ .Your answer should be in decimal form rounded to one decimal place.

Studdy Solution

STEP 1

What is this asking? We need to find the value of a limit as $x$ approaches 0 from the right, dealing with a tricky fraction involving an exponential. Watch out! Don't forget that $x$ is approaching 0 from the positive side, which is key for understanding what happens to $\frac{1}{x}$ .

STEP 2

1. Analyze the exponent
2. Evaluate the limit

STEP 3

Alright, let's break this down!
We have this expression $\frac{1}{x}$ , and we know $x$ is getting super close to 0, but from the positive side.

STEP 4

So, imagine $x$ as a tiny positive number like 0.001.
What happens to $\frac{1}{x}$ ?
It becomes HUGE!
Like, $\frac{1}{0.001} = 1000$ .
As $x$ gets even closer to 0, $\frac{1}{x}$ explodes towards positive infinity.
We write this as: $\lim_{x \rightarrow 0^{+}} \frac{1}{x} = \infty$

STEP 5

Now, what about $-\frac{1}{x}$ ?
Well, if $\frac{1}{x}$ goes to positive infinity, then $-\frac{1}{x}$ goes to negative infinity: $\lim_{x \rightarrow 0^{+}} -\frac{1}{x} = -\infty$

STEP 6

Let's look at $e^{-\frac{1}{x}}$ .
As we just saw, $-\frac{1}{x}$ goes to negative infinity.
What happens when the exponent in $e^x$ becomes super negative? $e$ to a very large negative power gets incredibly close to zero: $\lim_{x \rightarrow 0^{+}} e^{-\frac{1}{x}} = 0$

STEP 7

Now we can plug this back into our original expression: $\lim_{x \rightarrow 0^{+}} \frac{1}{2 + e^{-\frac{1}{x}}}$ Since $e^{-\frac{1}{x}}$ approaches 0, the whole expression becomes: $\lim_{x \rightarrow 0^{+}} \frac{1}{2 + e^{-\frac{1}{x}}} = \frac{1}{2 + 0}$

STEP 8

And finally, we calculate the result: $\frac{1}{2} = 0.5$

STEP 9

The limit is 0.5.

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QuestionQuestion 5
What is the value of $\lim _{x \rightarrow 0^{+}} \frac{1}{2+e^{-\frac{1}{x}}}$ .Your answer should be in decimal form rounded to one decimal place.

Studdy Solution

STEP 1

What is this asking? We need to find the value of a limit as $x$ approaches 0 from the right, dealing with a tricky fraction involving an exponential. Watch out! Don't forget that $x$ is approaching 0 from the positive side, which is key for understanding what happens to $\frac{1}{x}$ .

STEP 2

1. Analyze the exponent
2. Evaluate the limit

STEP 3

Alright, let's break this down!
We have this expression $\frac{1}{x}$ , and we know $x$ is getting super close to 0, but from the positive side.

STEP 4

So, imagine $x$ as a tiny positive number like 0.001.
What happens to $\frac{1}{x}$ ?
It becomes HUGE!
Like, $\frac{1}{0.001} = 1000$ .
As $x$ gets even closer to 0, $\frac{1}{x}$ explodes towards positive infinity.
We write this as: $\lim_{x \rightarrow 0^{+}} \frac{1}{x} = \infty$

STEP 5

Now, what about $-\frac{1}{x}$ ?
Well, if $\frac{1}{x}$ goes to positive infinity, then $-\frac{1}{x}$ goes to negative infinity: $\lim_{x \rightarrow 0^{+}} -\frac{1}{x} = -\infty$

STEP 6

Let's look at $e^{-\frac{1}{x}}$ .
As we just saw, $-\frac{1}{x}$ goes to negative infinity.
What happens when the exponent in $e^x$ becomes super negative? $e$ to a very large negative power gets incredibly close to zero: $\lim_{x \rightarrow 0^{+}} e^{-\frac{1}{x}} = 0$

STEP 7

Now we can plug this back into our original expression: $\lim_{x \rightarrow 0^{+}} \frac{1}{2 + e^{-\frac{1}{x}}}$ Since $e^{-\frac{1}{x}}$ approaches 0, the whole expression becomes: $\lim_{x \rightarrow 0^{+}} \frac{1}{2 + e^{-\frac{1}{x}}} = \frac{1}{2 + 0}$

STEP 8

And finally, we calculate the result: $\frac{1}{2} = 0.5$

STEP 9

The limit is 0.5.

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QuestionQuestion 5What is the value of lim⁡x→0+12+e−1x\lim _{x \rightarrow 0^{+}} \frac{1}{2+e^{-\frac{1}{x}}}limx→0+​2+e−x1​1​.Your answer should be in decimal form rounded to one decimal place.

Studdy Solution

STEP 1

What is this asking? We need to find the value of a limit as xxx approaches 0 from the right, dealing with a tricky fraction involving an exponential. Watch out! Don't forget that xxx is approaching 0 from the *positive* side, which is key for understanding what happens to 1x\frac{1}{x}x1​.

STEP 2

1. Analyze the exponent2. Evaluate the limit

STEP 3

Alright, let's **break this down**!We have this expression 1x\frac{1}{x}x1​, and we know xxx is getting super close to 0, but from the *positive* side.

STEP 4

STEP 5

Now, what about −1x-\frac{1}{x}−x1​?Well, if 1x\frac{1}{x}x1​ goes to positive infinity, then −1x-\frac{1}{x}−x1​ goes to **negative infinity**: lim⁡x→0+−1x=−∞ \lim_{x \rightarrow 0^{+}} -\frac{1}{x} = -\infty x→0+lim​−x1​=−∞

STEP 6

STEP 7

STEP 8

And finally, we **calculate the result**: 12=0.5 \frac{1}{2} = 0.5 21​=0.5

STEP 9

The limit is **0.5**.

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QuestionQuestion 5
What is the value of $\lim _{x \rightarrow 0^{+}} \frac{1}{2+e^{-\frac{1}{x}}}$ .Your answer should be in decimal form rounded to one decimal place.

What is this asking? We need to find the value of a limit as $x$ approaches 0 from the right, dealing with a tricky fraction involving an exponential. Watch out! Don't forget that $x$ is approaching 0 from the positive side, which is key for understanding what happens to $\frac{1}{x}$ .

1. Analyze the exponent
2. Evaluate the limit

Alright, let's break this down!
We have this expression $\frac{1}{x}$ , and we know $x$ is getting super close to 0, but from the positive side.

Now, what about $-\frac{1}{x}$ ?
Well, if $\frac{1}{x}$ goes to positive infinity, then $-\frac{1}{x}$ goes to negative infinity: $\lim_{x \rightarrow 0^{+}} -\frac{1}{x} = -\infty$

And finally, we calculate the result: $\frac{1}{2} = 0.5$

The limit is 0.5.