The value of constant c for which the function k(x) is continuous is zero.
What is the limit of a function?
The limit of a function at a point k in its field is the value that the function approaches as its parameter approaches k.
To determine the value of constant c for which the function of k(x) is continuous, we take the limit of the parameter as follows:
[tex]\mathbf{ \lim_{x \to 0^-} k(x) = \lim_{x \to 0^+} k(x) = 0 }[/tex]
[tex]\mathbf{\implies \lim_{x \to 0 } \ \ \dfrac{sec \ x - 1}{x}= c }[/tex]
Provided that:
[tex]\mathbf{\implies \lim_{x \to 0 } \ \ \dfrac{sec \ x - 1}{x}= \dfrac{0}{0} \ (form) }[/tex]
Using l'Hospital's rule:
[tex]\mathbf{\implies \lim_{x \to 0} \ \ \dfrac{\dfrac{d}{dx}(sec \ x - 1)}{\dfrac{d}{dx}(x)}= \lim_{x \to 0} sec \ x \ tan \ x = 0}[/tex]
Therefore:
[tex]\mathbf{\implies \lim_{x \to 0 } \ \ \dfrac{sec \ x - 1}{x}=0 }[/tex]
Hence; c = 0
Learn more about the limit of a function x here:
https://brainly.com/question/8131777
#SPJ1
