Limits and exponential function

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I'm having trouble with this question on my homework for Calc I, I was hoping I could get some help here.

[latent jolt]

You want the numerator to “outpace” the denominator

Therefore you need the denominator to essentially go to 0, stop growing, grow slower than the numerator, or help the numerator grow actually grow

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im new to limits so I'm trying to understand. Because I tried to input 0 as a value for k and its said it was incorrect

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how should I approach this problem numerically?

should I just start assigning numbers? or just go by how it looks the equation will behave

[latent jolt]

I’m attempting to read the question again and I’m unsure if the limit approaches infinity if that means the limit exists

Ok I think it means that the limit should approach some constant possibly

Give me a moment

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Thank you so much . I've been starring at this question for about an hour by now lol

[latent jolt]

Ok so for k values equal to or greater than ln2 will make the limit equal a constant which should make it exist

According to Google limits TECHNICALLY do not exist when they equal infinity so we are looking for e^kx > 2^x therefore the limit will not approach infinity

And will approach a constant

limit = 1 when k = ln2
limit = infinity (DNE) when k < ln2
limit = 0 when k > ln2

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omg thank you so much I think I understand it better know and Just input the ln2 in the homework and It work. Thank you so much! I really appreciate your help! You're a life saver 🥹

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Thank you so much really appreciate your help 🥹

[latent jolt]

Actually ignore my thought about greater than Ln2 means their is a limit I am unsure but u got it correct so I guess it works

e^kx = 2^x

kx=xln(2)

k= ln(2)

In case u wanted the work

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Thanks again🥹 (thank you for the work I'll check it tomorrow when my brain is fully awake lol)

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.solved