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Before we jump into derivatives, we need to talk about limits. A *limit* is essentially what happens to a function as it approaches a certain value. It's crucial for understanding continuity and derivatives. Think of it like this: imagine walking towards a door. The limit is where you're headed, even if you never actually reach the door. A function is **continuous** if you can draw its graph without lifting your pen. Mathematically, this means the limit of the function as x approaches a point 'c' exists, the function is defined at 'c', and the limit equals the function's value at 'c'. In simpler terms, there are no breaks, jumps, or holes in the graph at that point. Understanding continuity is vital because many theorems in calculus only apply to continuous functions. For example, the Intermediate Value Theorem guarantees that if a continuous function takes on two values, it must also take on every value in between. Mastering limits and continuity is not just about memorizing definitions; it’s about understanding how functions behave and setting the stage for derivatives and integrals.
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We also need to consider emerging markets and how they could affect Sidney's economy. The rise of new technologies and industries will also play a critical role. For instance, **_investments in renewable energy, artificial intelligence (AI), and biotechnology_** could significantly boost economic growth.
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* **Streaming Services**: Platforms for online viewing.
