Question

ha (predict the tuto por certain town 5. A function fis said to be continuous at x = a if f(a) is defined, if lim f(x) exists, and if lim/(x) = f(a). Use this definition to explain why the following function is not continuous at x = 2. Include in your explanation an accurate sketch of this function. Also be sure to state specific values of "a", lim f(x), and f(a) in your response. Slx - 21 if x # 2 3 if x = 2 f(x) = {lx 30. Find the taste the

Answer 1

Here, a (6,4) 3 (2,3) 2 a with values | 20-21 ir xt2 foxs = 3 if x=2 The sketch of f is given by, y-ancis 4 (-2,4) Co; 2) (4,2) 2-asis -4 - 3 4 5 6 Sketch of Here from above graph we can see that lim f(x) = lim 1 20-21 06-27 - 2- MAR lim -(x-2) 06-)2-" ('lim xa of x <a =) 26-aco =) 12-a1 = -(2-a) + t -2 2 سر f こ ..lim x-) 2- fux) - (2-2) = - CO =O-(i)

Also, lim 142t fory lim 24 2+ 12-21 = lim (2-2) jj2+ means approaching a with values lim - at fesy of x > a - a > 0 =) le-al = (2-a) =) lim (-) 2 + fores (2-2)=0 ) ۔ So, from co and (ii) carol lim X-2 fex) O Also, too from Sketch of f we can see that f(2)=3 So, lim fesc) * = 0 + f(2) = 3 2-)2 f is not continuous So, By given given definition, at X = 2 ie, f is discontinuous at = 2.