Find the limit $\lim_{k\rightarrow\infty}k\int_{[1-\frac{1}{k}, 1]}f(x) \text{d}x$ when $\text{spt}(f)\subset [0, 2]$

Question

I have the following problem. Find the limit $$\lim_{k\rightarrow\infty}k\int_{[1-\frac{1}{k}, 1]}f(x) \text{d}x$$ if it exists, when $f$ is continuous and has compact support in $[0, 2]$. I tried dominated convergence but got stuck because the function $kf(x)$ tends to infinity and the interval tends to single point $\{1\}$ and thus the integral is zero. But that integral looks like integrating w.r.t. Dirac delta function so I guess the answer should be $f(1)$. Can someone help me with this problem?

Answer 1

Observe that: $$k\int_{[1-\frac{1}{k},1]}f(1)= f(1)$$ Now, can use the continuity of $f(x)$ at $1$?

Answer 2

If $f(x)$ is continuous in $\subset [0, 2]$, it exist some $F(x)=\int f(x) \text{d}x$, continuous an derivable in the same region. So by Barrow we have

$$I=\lim_{k\rightarrow\infty}k\int_{[1-\frac{1}{k}, 1]}f(x) \text{d}x= =\lim_{k\rightarrow\infty}k [F(1)-F(1-\frac{1}{k})]$$

Now as $F(x)$ is derivable with continuity at $x=1$, we can expand it in some neighbourhood by Taylor $$F(1-\frac{1}{k})=F(1)-\frac{1}{k}F'(1)+O(x^2)=F(1)-\frac{1}{k}f(1)+O(\frac{1}{k^2})$$

So by substituing

$$I=\lim_{k\rightarrow\infty}k [F(1)-F(1)+\frac{1}{k}f(1)-O(\frac{1}{k^2})] =f(1)-\lim_{k\rightarrow\infty}k O(\frac{1}{k^2})=f(1)$$