Main Question [5 marks]

Meaning of $\lim_{x \to 2} f(x) = 5$

Definition: The statement $\lim_{x \to 2} f(x) = 5$ means that as $x$ gets closer and closer to 2 (from both left and right sides), the value of $f(x)$ gets closer and closer to 5, regardless of the actual value of $f$ at $x = 2$.

Key Points:

• The limit describes the behaviour of the function near the point, not at the point.
• $x$ approaches 2 but never actually equals 2.
• For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - 2| < \delta$, then $|f(x) - 5| < \epsilon$.

Is it possible that $\lim_{x \to 2} f(x) = 5$ yet $f(2) = 3$?

Yes, it is absolutely possible.

The limit of a function as $x \to 2$ does not depend on the value of the function at $x = 2$. The limit only concerns values of $f(x)$ when $x$ is near 2, not equal to 2. So the function can have a limit of 5 as $x$ approaches 2, while the actual value $f(2)$ is defined as 3 (or any other number, or even undefined).

This situation occurs when there is a removable discontinuity (a hole) in the graph at $x = 2$, where the graph approaches $y = 5$ but the point plotted is at $(2, 3)$.

Conclusion: A limit at a point and the function value at that point are two independent concepts — they need not be equal.

---

Sub-question 1 [5 marks]

Graph, Domain, and Range of $f(x) = x^2 + 4$

Definition: $f(x) = x^2 + 4$ is an upward-opening parabola shifted 4 units up from the origin.

Graph Description

Domain:

• $f(x) = x^2 + 4$ is defined for all real values of $x$.

$$\text{Domain} = (-\infty, \infty) \text{ or } \mathbb{R}$$

Range:

• Since $x^2 \geq 0$ for all real $x$, the minimum value of $f(x) = x^2 + 4$ is $0 + 4 = 4$.
• The function can grow infinitely large but never goes below 4.

$$\text{Range} = [4, \infty)$$

Conclusion: $f(x) = x^2 + 4$ is a continuous parabola with vertex at $(0, 4)$, defined for all real numbers, with output values always greater than or equal to 4.