Question

The following limit represents for some function f and some real number a.

a. Find a possible function f and number a.
b. Evaluate the limit by computing.

Answer 1

a.
`\( \lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}} \)` represents the function f(x) = x + 2 and the number a = 2.

b. The limit evaluates to 4.

Step-by-step explanation:

For the given limit
`\( \lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}} \)`, we can simplify the expression by factoring the numerator as the difference of squares:
`\( \frac{{(x + 2)(x - 2)}}{{x - 2}} \)`. Canceling out the common factor of ( x - 2 ), we get ( x + 2 ), which is the function f(x) for this limit. The value of a corresponds to the number that x is approaching, which in this case is 2.

By directly substituting ( x = 2 ) into the function f(x) = x + 2 , we find the limit's value to be f(2) = 2 + 2 = 4. This result demonstrates that as x approaches 2, the function f(x) = x + 2 approaches 4, which is the value of the limit.