Answered: "How do you prove that the function f..."

College
Calculus and Analysis
Differential Calculus
Continuity

Zachariah Norris

Answered question

2022-09-15

How do you prove that the function

f (x) = x^{2} - 3 x + 5

is continuous at a =2?

Answer & Explanation

ti1k1le2l

Beginner2022-09-16Added 9 answers

Any polynomial function is continuous everywhere, but let's prove this particular example using the limit definition of continuity...
A function f(x) is continuous at a point a if both f(a) and

lim_{x \to a} f (x)

are defined and equal.
In our example:

f (2) = 2^{2} - 3 (2) + 5 = 4 - 6 + 5 = 3

= 2^{2} + 4 h + h^{2} - 3 (2) - 3 h + 5

= 2^{2} + 4 h + h^{2} - 3 (2) - 3 h + 5

= 4 - 6 + 5 + h + h^{2}

= 3 + h + h^{2} \to 3

h \to 0

lim_{x \to 2} f (x) = lim_{h \to 0} f (2 + h) = 3 = f (2)

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