1. ## Limits at Infinity

Evaluate the limit at infinity. f(x) = 4[(x + sin x)/(x)]

lim x→∞ f(x)

The limit of sinx as x approaches infinity = does not exist, which leaves me with x/x or ∞/∞ = 1 so 4(1)=4?

Thanks in advance for the help!

2. ## Re: Limits at Infinity

Originally Posted by Johnny Walker Black
Evaluate the limit at infinity. f(x) = 4[(x + sin x)/(x)]

lim x→∞ f(x)

The limit of sinx as x approaches infinity = does not exist, which leaves me with x/x or ∞/∞ = 1 so 4(1)=4?

Thanks in advance for the help!
For large x:
$\displaystyle x >> |\sin(x)|$

Therefore your sin(x) term is negligible

Edit: Visualise by thinking of a suitably large number - say $\displaystyle 10^6$ then $\displaystyle \sin(x) \approx -0.35$.

Now what is $\displaystyle 10^6 - (-0.35)$

3. ## Re: Limits at Infinity

Originally Posted by Johnny Walker Black
Evaluate the limit at infinity. f(x) = 4[(x + sin x)/(x)]

lim x→∞ f(x)

The limit of sinx as x approaches infinity = does not exist, which leaves me with x/x or ∞/∞ = 1 so 4(1)=4?

Thanks in advance for the help!
$\displaystyle f(x)= 4\frac{x+ sin(x)}{x}= 4(1+ \frac{sin(x)}{x})$
Now, $\displaystyle -1\le sin(x)\le 1$ for all x so as the denominator goes to infinity, $\displaystyle \frac{sin(x)}{x}$ goes to 0.