Math

QuestionFind the limit as xx approaches 3 for the expression x1/2(5x7)1/3x^{-1/2}(5x-7)^{1/3}.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as xx approaches 33 for the function x1/(5x7)1/3x^{-1 /}(5 x-7)^{1 /3}. . We are assuming that the function is continuous at the point x=3x =3.

STEP 2

First, let's substitute x=x = into the function to see if we can directly evaluate the limit.
limx[x1/2(5x7)1/]=1/2(57)1/\lim{x \rightarrow}\left[x^{-1 /2}(5 x-7)^{1 /}\right] =^{-1 /2}(5 \cdot-7)^{1 /}

STEP 3

implify the expression inside the limit.
=31(157)1/3= \sqrt{3}^{-1}(15-7)^{1 /3}

STEP 4

Further simplify the expression.
=31(8)1/3= \sqrt{3}^{-1}(8)^{1 /3}

STEP 5

Evaluate the cube root and the reciprocal of the square root.
=132= \frac{1}{\sqrt{3}} \cdot2

STEP 6

implify the expression to get the final result.
=23= \frac{2}{\sqrt{3}}So, limx3[x1/2(5x)1/3]=23\lim{x \rightarrow3}\left[x^{-1 /2}(5 x-)^{1 /3}\right] = \frac{2}{\sqrt{3}}.

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