I don't understand how to calculate the limit $$\lim_{x\to7}\frac{1}{x-7}\int_7^x\sqrt{t^2+9}dt $$ without using the L'Hopital rule the picture.
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2Please do not use pictures. – Dietrich Burde Jun 01 '19 at 16:57
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It shuold be $$2\sqrt{14}$$ – Dr. Sonnhard Graubner Jun 01 '19 at 17:03
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Welcome to Math StackExchange. Please show what you have tried - this is not a site meant to have people solve homework for you. Also, please use [MathJax][1] to present the formulae equations. [1]: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Adam Latosiński Jun 01 '19 at 17:03
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If you don't want to use l'Hopital, you'll be in trouble with that limit. – Crostul Jun 01 '19 at 17:03
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For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – Martin Sleziak Jun 03 '19 at 10:03
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Using $f(t)=\sqrt{t^2+9}$ $$\int_7^x f(t)dt=F(x)-F(7)$$ Then what you are asked to compute is $$\lim_{x\to 7}\frac{F(x)-F(7)}{x-7}=F'(x)|_{x=7}=f(7)$$
Andrei
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Hint: use mean value theorem. There exists $\xi \in [7,x]$ such that $$\int_7^x \sqrt{t^2+9}dt = (x-7) \sqrt{\xi^2+9}$$ You don't really need to know the value of $\xi$, the fact that $\xi \in [7,x]$ is enough.
Adam Latosiński
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