The function f(x) defined as (sincx/x) for x<0 and 1+(c)(tan2x/x) for x≥0 requires the value of c to ensure continuity at x=0. The limit as x approaches 0 from the right yields 1 + 2c, while the limit from the left results in c. Setting these two limits equal leads to the equation 1 + 2c = c, which simplifies to c = -1. Therefore, for the function to be continuous at x=0, the definitive value of c is -1.
PREREQUISITESStudents studying calculus, particularly those focusing on limits and continuity, as well as educators looking for examples of piecewise functions in mathematical analysis.
Looks good to me.AlexandraMarie112 said:Homework Statement
f(x)= (sincx/x) ; x<0
1+(c)(tan2x/x) ; x≥0
Homework Equations
The Attempt at a Solution
Lim as x tends to 0+[/B] = 1+c⋅2⋅(sin2x/2x)⋅(1/cos2x) =1+c⋅2⋅1⋅1= 1+2c
Lim as x tends to 0 - = (sincx/x)=(c/1)⋅(sinx/x)=c⋅1=c
Equating both: 1+2c=c
c=-1
∴For the function to be continuous at x=0, c=-1
Is my answer correct?