Nth root as value inside the root symbol is less than 1L=limx→0+xa=0(a&gt;0)

Step 1For any fixed positive a the function given byf(x)=xais a continuous increasing function of x, with f(0)=0. One way to see that is to note that its derivativef′(x)=axa−1&gt;0for x&gt;0The fact that a−1&lt;0 when a&lt;1 means that the graph of f is tangent (from the right) to the y-axis at the origin. So the slope is badly behaved there, but the function itself has limit 0.Look at the graph of y=x to see what is going on.

Question: "Nth root as value inside the root symbol..."

High School
Algebra
Pre-Algebra
Decimals

Jaidyn Cook

Answered question

2022-04-23

Nth root as value inside the root symbol is less than 1

L = lim_{x \to 0^{+}} x^{a} = 0

(a > 0)

Answer & Explanation

eslasadanv3

Beginner2022-04-24Added 20 answers

Step 1
For any fixed positive a the function given by

f (x) = x^{a}

is a continuous increasing function of x, with

f (0) = 0

. One way to see that is to note that its derivative

f^{'} (x) = a x^{a - 1} > 0

for

x > 0

The fact that

a - 1 < 0

when

a < 1

means that the graph of f is tangent (from the right) to the y-axis at the origin. So the slope is badly behaved there, but the function itself has limit 0.
Look at the graph of

y = \sqrt{x}

to see what is going on.

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