query: Suppose limit as x approaches 0 of the derivative of f(x)=a. Show that the derivative at 0 exists and it is equal to a.
positive_passages.docid: 25592
positive_passages.text: If a derivative of a continuous function has a limit, must it agree with that limit?
negative_passages.docid: 36358
negative_passages.text: How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?
negative_passages.docid: 28949
negative_passages.text: Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
negative_passages.docid: 101778
negative_passages.text: Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
negative_passages.docid: 122762
negative_passages.text: Proving that the limit doesn't exist for $x$ other than $0$
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 154321
negative_passages.text: How to show that a limit diverges to $−∞$
negative_passages.docid: 112898
negative_passages.text: Derivative of sin(x)/x at $0$ by definition of derivative
negative_passages.docid: 41037
negative_passages.text: Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
negative_passages.docid: 145496
negative_passages.text: Show $\lim_{h \to \ 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x)$ Proof verification
negative_passages.docid: 13150
negative_passages.text: If the derivative of $f$ is never zero, then $f$ is one-to-one
negative_passages.docid: 151958
negative_passages.text: How to show $f'(0)$ exist and is equal to $1$?
negative_passages.docid: 44370
negative_passages.text: If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
negative_passages.docid: 30204
negative_passages.text: Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
negative_passages.docid: 119907
negative_passages.text: Finding the Derivative of |x| using the Limit Definition
negative_passages.docid: 143874
negative_passages.text: Limit of derivative is zero. Does it imply a limit for f(x)?
negative_passages.docid: 106867
negative_passages.text: Limit of $\cos(x)/x$ as $x$ approaches $0$