(a) (i) Let f: JR be a real function defined on the non-empty open interval JC...

Transcribed Image Text:

(a) (i) Let f: JR be a real function defined on the non-empty open interval JC R. Define what it means for f to have limit L at the point a € J. (ii) Using this definition, show that the function f: RR given by f(x)= x² + 2x + 1 satisfies lim f(x) = 9. x-2 (b) (i) State the Intermediate Value Theorem. (ii) - Prove that the equation x² + x = cos x = 0 has a solution in [0, 1]. (iii) Why must this solution be unique? (a) (i) Let f: JR be a real function defined on the non-empty open interval JC R. Define what it means for f to have limit L at the point a € J. (ii) Using this definition, show that the function f: RR given by f(x)= x² + 2x + 1 satisfies lim f(x) = 9. x-2 (b) (i) State the Intermediate Value Theorem. (ii) - Prove that the equation x² + x = cos x = 0 has a solution in [0, 1]. (iii) Why must this solution be unique?