Limits of a Function: Find e1 & -Infinite

[masterchiefo]

Homework Statement

1. f(x) = (1+x)^1/x
lim _x->0⁺ (1+x)^1/x

2. lim _x->0⁺ 10+ln(x)

Homework Equations

The Attempt at a Solution

1. lim _x->0⁺ (1+x)^1/x

(1+0⁺)^1/0+

I really don't understand how the final answer is e... thanks for helping.

= e¹2. lim _x->0⁺ 10+ln(x)

ln(0+) = -infinite
10 - infinite = - infinite
I don't understand why ln(x) is - infinite is there any rule for that or that's something I have to know by heart ?

 

[Svein]

1. If you let u=\frac{1}{x}, then \lim_{x\rightarrow 0+} (1+x)^{\frac{1}{x}} transforms into \lim_{u\rightarrow +\infty}(1+\frac{1}{u})^{u}. Does this expression look familiar? (http://en.wikipedia.org/wiki/E_(mathematical_constant)).
2. Do the same transformation here ( u=\frac{1}{x}). The expression changes to \lim_{u\rightarrow +\infty}(10+ln(\frac{1}{u}))=10+\lim_{u\rightarrow +\infty}ln(\frac{1}{u})=10-\lim_{u\rightarrow +\infty}ln(u).

 

[SammyS]

Homework Statement

1. f(x) = (1+x)^1/x
lim _x->0⁺ (1+x)^1/x

2. lim _x->0⁺ 10+ln(x)

Homework Equations

The Attempt at a Solution

1. lim _x->0⁺ (1+x)^1/x

(1+0⁺)^1/0+

I really don't understand how the final answer is e... thanks for helping.

= e¹2. lim _x->0⁺ 10+ln(x)

ln(0+) = -infinite
10 - infinite = - infinite
I don't understand why ln(x) is - infinite is there any rule for that or that's something I have to know by heart ?

For item #2.

I don't know whether or not you want to consider it as " knowing by heart " , but if you're serious about math, physics, engineering, etc., you should be very familiar with the characteristics of logarithmic functions.

ln(x) is strictly increasing, it's domain is the interval $\displaystyle \ (0,\ \infty) \$, it's range is $\displaystyle \ (-\infty,\ \infty) \$.

That's enough to get $\displaystyle \ \lim_{x\to0^+} \ln(x)=-\infty \$.

I or someone else will cover item #1 in a separate post.

 

[nuuskur]

$x = e^{\ln x}$. In our current problem, $x$ represents the expression your are taking the limit of.
In the process $x\to 0$ , the expression $\ln (1+x)$ is equivalent to $x$.

E: This is basic mathematical analysis and not about physics. Not being mean, just pointing it out.

Also, usually $x= e^{\ln |x|}$, however here we approach $0$ strictly from the right side hence the $x$ is positive regardless.

 

[vela]

I don't understand why ln(x) is -infinity. Is there any rule for that or is that something I have to know by heart?

You should be able to reason it out. The relation $y = \ln x$ is equivalent to $x = e^y$. If you want to make x very very small, i.e. close to 0, what kind of value do you have to use for y?