Trying to compute a limit for the Turán number

Question

I have been trying to figure out how to compute the limit shown below.

Let me replace $p-1$ with $k$ to make things easier. If $v$ is a multiple of $k$, that is, $v=qk$ for some integer $q$, then I can compute this limit with no problem. My issue is when $v=qk+r$, where $1\leq r <k$. I get a long expression and I cannot see how to compute the answer.

Answer 1

The expressions are not so long and they collapse

$$\lim_{v\to\infty}\frac{e(T_{v,k})}{v^2}=$$

$$\lim_{v\to\infty}\frac{{r\choose 2}(q+1)^2+{k-r\choose 2}q^2+r(k-r)q(q+1)}{v^2}=$$

$$\lim_{v\to\infty}\frac{r(r-1)(q+1)^2+(k-r)(k-r-1) q^2+2r(k-r)q(q+1)}{2v^2}=$$

$$\lim_{v\to\infty}\frac{q^2k^2-q^2k+2qkr-2rq+r^2-r}{2v^2}=$$

$$\lim_{v\to\infty}\frac{(v-r)^2-(v-r)^2/k+2(v-r)r-2r(v-r)/k+r^2-r}{2v^2}=$$

$$\lim_{v\to\infty}\frac{kv^2-v^2+r^2-rk}{2kv^2}=$$

$$\lim_{v\to\infty}\frac 12-\frac 1{2k}+\frac{r^2-rk}{2kv^2}=$$

$$\frac 12-\frac 1{2k}.$$