Question

Explain in your own words what is meant by the equation limₓ→₂ f(x) = 5. Is it possible for this statement to be true and yet f(2) = 3?

Answer 1

Final answer:

The equation
`\(\lim_{{x \to 2}} f(x) = 5\)` indicates that as
`\(x\)` approaches 2, the function values approach 5; it is possible for this limit to be true while
`\(f(2)\)` takes a different value, such as
`\(f(2) = 3\)`.

Step-by-step explanation:

The equation
`\(\lim_{{x \to 2}} f(x) = 5\)` represents the limit of the function
`\(f(x)\)` as
`\(x\)` approaches 2. It implies that as
`\(x\)` gets arbitrarily close to 2 from both sides, the function values approach 5.

However, the statement
`\(\lim_{{x \to 2}} f(x) = 5\)` does not necessarily dictate the value of
`\(f(2)\)`. The limit only describes the behavior of the function as
`\(x\)` approaches 2, not its actual value at
`\(x = 2\)`.

So, it is entirely possible for the limit to be true
`(\(\lim_{{x \to 2}} f(x) = 5\))` while
`\(f(2)\)` takes a different value, such as
`\(f(2) = 3\)`. The limit provides information about the trend of the function near
`\(x = 2\)` but doesn't guarantee the value of the function at that specific point. The limit and the value at a particular point are distinct concepts in calculus.