If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning

Sum

`lim_(x -> 2)` = 4

`lim_(x -> 2^-) f(x) = lim_(x -> 2^+) f(x)` = 4

When x approaches 2 from the left or from the right f(x) approaches 4.

Given that `lim_(x -> 2^-) (x) = lim_(x -> 2^+)f(x)` = 4

The existence or non-existence at x = 2 has no leaving on the existence of the limit of f(x) as x approaches to 2.

∴ We cannot conclude the value of f(2).

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Chapter 9: Differential Calculus - Limits and Continuity - Exercise 9.1 [Page 98]

Chapter 9 Differential Calculus - Limits and Continuity
Exercise 9.1 | Q 21 | Page 98

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> 2)(2x + 3)` = 7

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> -3) (3x + 2)` = – 7

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 2) (x - 2)/(x^2 - 4)`

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	0.25641	0.25062	0.250062	0.24993	0.24937	0.24390

Sketch the graph of a function f that satisfies the given value:

f(– 2) = 0

f(2) = 0

`lim_(x -> 2) f(x)` = 0

`lim_(x -> 2) f(x)` does not exist.

Verify the existence of `lim_(x -> 1) f(x)`, where `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`

Evaluate the following limits:

`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) ("a" > "b")`

Evaluate the following limits:

`lim_(x -> 0)(1 + x)^(1/(3x))`

Evaluate the following limits:

`lim_(x -> oo)(1 + "k"/x)^("m"/x)`

Evaluate the following limits:

`lim_(x -> 0) (sin("a" + x) - sin("a" - x))/x`

Evaluate the following limits:

`lim_(x -> 0) (1 - cos^2x)/(x sin2x)`

Evaluate the following limits:

`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`