True or false: $\limsup_{x\rightarrow\infty}\left((x \mod 5)+\frac{1}{x+1} \right )=5$

Question

I'm not really sure because

$(x \mod 5) \leq4$

$\Rightarrow$

$4+\frac{1}{0+1}=4+1=5$

The statement would be true but we also need to know that it could be lower than $5$...

Im confused if statement is true or false now.

I think statement wrong then because limes sup goes to $\infty$ so modulo won't be 4, it will be very very close to 4. So final result will be almost 5 but not exactly (it will be lower)?

Answer 1

Hint: Take $x_n=5n-\frac{1}{n}$ and calculate the limit for this sequence.