How to Find Constants for a Continuous Function?

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SUMMARY

The discussion focuses on finding constants a and b to ensure the continuity of the function g(x) defined as g(x)={4 sin(x)/x, x<0; a-2x, x≥0}. The key point is addressing the potential discontinuity at x=0, where the limit of sin(x)/x as x approaches 0 is established as 1. Therefore, for continuity, it is necessary that a equals 2, ensuring that both sides of the function match at x=0.

PREREQUISITES
  • Understanding of limits in calculus
  • Knowledge of continuity in functions
  • Familiarity with piecewise functions
  • Basic trigonometric limits, specifically lim (sin x)/x as x approaches 0
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  • Study the concept of limits in calculus, focusing on trigonometric limits
  • Learn about continuity and discontinuity in piecewise functions
  • Explore the application of the epsilon-delta definition of continuity
  • Practice solving problems involving piecewise-defined functions
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Students in calculus, particularly those studying limits and continuity, as well as educators looking for examples of piecewise functions in mathematical analysis.

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Homework Statement


Find the constants a and b such that the function is continuous on the entire line.


Homework Equations


g(x)={4 sinx/x, x<0
{a-2x, x> or = 0


The Attempt at a Solution


Possible discontinuity at x=0
f(0^+)=a-2x=a-2(0)=a
f(0^-)=4sinx/x=4sin(0)/(0)

i am stuck, wouldn't f(0^-) not exist then? i really don't know what I am doing on this entire assignment as i have been gone from school for a week so if anyone knows how to do this, I would love for you to help me on AIM or MSN tonight.
 
Physics news on Phys.org
You seem to be doing fine, except you didn't know that the limit of sinx/x as x-->0 is 1. Now you know!
 

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