Question

suppose that f has a positive derivative for all values of x

Answer 1

We have that f(x) exists described for all real values of x, except for $x=x _0$. $\lim _{x \rightarrow x 0} f(x)$

What is meant by positive derivative?

The graph shows a rising trend when the derivative's sign is positive. In all cases where x > 0, the derivative's sign is positive.

A function is increasing, decreasing, or constant on an interval if the derivative is positive, negative, or zero on that interval.

It is possible to determine the slope of a tangent line to a curve at any time using a function's first derivative. The first derivative of a function provides us with a wealth of information about the function as a result of this definition. Obviously increasing if is positive. is decreasing if it is negative.

Remember that when we are taking the limit we are not evaluating the function in $x_0$, instead, we are evaluating the function in values really close to $x_0$ (values defined as $\mathrm{xO}^{+}$and $\mathrm{xO}^{-}$, where the sign defines if we approach from above or bellow).

And because f(x) is defined in the values of x near $x_0$, we can conclude that the limit does exist if:

$\lim _{x \rightarrow 0+} f(x)=\lim _{x \rightarrow 0-} f(x)$

if that does not happen, like in f(x) = 1 / x where $x_0=0$

where the lower limit is negative and the upper limit is positive, we have that the limit does not converge.

The complete question is:

Suppose that a function f(x) is defined for all real values of x, except x = xo. Can anything be said about LaTeX: \displaystyle\lim\limits_{x\to x_0} f(x)lim x → x 0 f ( x )? Give reasons for your answer.

To learn more about derivatives refer to:

https://brainly.com/question/28376218

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