Computing limits using u substitution

[lbrizzee]

3. When computing the limit of a composition of functions, we can introduce a substitution. For example, consider:

. . . . .\(\displaystyle \displaystyle \lim_{x\, \rightarrow \, \infty}\, \ln\left(\dfrac{2}{x^3}\right)\)

Let \(\displaystyle \, u\, =\, \dfrac{2}{x^3}\, \) and note that, as \(\displaystyle \, x\, \rightarrow \, \infty,\, u\, \rightarrow \, 0^+.\,\) Thus, we can rewrite the limit (in terms of u only), as:

. . . . .\(\displaystyle \displaystyle \lim_{u\, \rightarrow \, 0^+}\, \ln(u)\)

a. What is the value of this limit?

b. Explain why u approaches 0 specifically from the right (hence the use of 0⁺).

c. Using this method, compute \(\displaystyle \, \displaystyle \lim_{\theta \, \rightarrow \, 0^-}\, 2^{\cot \left(\theta\right)}\)

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I've attached a picture of my calculus question to this thread, although I'm not sure if it worked or if it is visible. If it worked, I would greatly appreciate some help!! I only need help with parts a and c. If someone could explain the steps, that would be wonderful!

 

[ksdhart]

3. When computing the limit of a composition of functions, we can introduce a substitution. For example, consider:

. . . . .\(\displaystyle \displaystyle \lim_{x\, \rightarrow \, \infty}\, \ln\left(\dfrac{2}{x^3}\right)\)

Let \(\displaystyle \, u\, =\, \dfrac{2}{x^3}\, \) and note that, as \(\displaystyle \, x\, \rightarrow \, \infty,\, u\, \rightarrow \, 0^+.\,\) Thus, we can rewrite the limit (in terms of u only), as:

. . . . .\(\displaystyle \displaystyle \lim_{u\, \rightarrow \, 0^+}\, \ln(u)\)

a. What is the value of this limit?

b. Explain why u approaches 0 specifically from the right (hence the use of 0⁺).

c. Using this method, compute \(\displaystyle \, \displaystyle \lim_{\theta \, \rightarrow \, 0^-}\, 2^{\cot \left(\theta\right)}\)

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I've attached a picture of my calculus question to this thread, although I'm not sure if it worked or if it is visible. If it worked, I would greatly appreciate some help!! I only need help with parts a and c. If someone could explain the steps, that would be wonderful!

Alright, so, you're having trouble with parts A and C. Let's start with part A. What do you know about limits? What do they represent? Since u approaches 0 from the right, what does that mean about the value of u​? And what happens to the value of ln(u)? Is it approaching some value?

And for part C, you have a limit as the angle theta approaches 0. But note that it says 0^-. If 0⁺ signifies a right-hand limit, then what does 0^- signify? What does that mean about the values of theta? As theta gets closer and closer to 0, what is the value of \(\displaystyle cot\left(\theta \right)\)? And what is the value of \(\displaystyle 2^{cot\left(\theta \right)}\)?