91Ó°ÊÓ

A continuous function is a function that has no breaks, jumps, or holes in its graph. If you can draw the function's graph without lifting your pen, then the function is continuous. Mathematically, a function \( f(x) \) is continuous at a point \( x = c \) if the following conditions are met:

• The function is defined at that point, \( f(c) \) exists.
• The limit as \( x \) approaches \( c \) from the left, \( \text{lim}_{x \to c^-} f(x) \), exists.
• The limit as \( x \) approaches \( c \) from the right, \( \text{lim}_{x \to c^+} f(x) \), exists.
• The value of the function at \( c \) equals the left-hand limit and the right-hand limit: \( f(c) = \text{lim}_{x \to c^-} f(x) = \text{lim}_{x \to c^+} f(x) \).

In simpler terms, you must be able to approach the point from both sides and get to the same function value.

Limits are fundamental in calculus and play a critical role in defining derivatives, integrals, and continuity. The limit of a function \( f(x) \) as \( x \) approaches a specific value \( c \) is written as \[ \text{lim}\thinspace_{x \to c} f(x) \]. This represents the value that \( f(x) \) gets closer to as \( x \) gets closer to \( c \). Here are some key points about limits:

• Right-Hand Limit: The limit as \( x \) approaches \( c \) from the positive side, noted as \[ \text{lim}\thinspace_{x \to c^+} f(x) \].
• Left-Hand Limit: The limit as \( x \) approaches \( c \) from the negative side, noted as \[ \text{lim}\thinspace_{x \to c^-} f(x) \].

In cases where the right-hand limit and the left-hand limit are equal, we say that the limit \( \text{lim}\thinspace_{x \to c} f(x) \) exists, and it is equal to that common value. Understanding limits is vital for studying continuity and other advanced concepts in calculus.

To determine the value that makes a function continuous at a certain point, you often need to evaluate its limits. Let's break down how we apply this to the provided exercise:

Right-Hand Limit
We need to find the limit of the function \( f(x) \) as \( x \) approaches 0 from the positive side:

\[ \text{lim}\thinspace_{x \to 0^+} 2(x - a) = 2(0 - a) = -2a \]

Left-Hand Limit
Next, we find the limit of the function \( f(x) \) as \( x \) approaches 0 from the negative side:

\[ \text{lim}\thinspace_{x \to 0^-} (x^2 + 1) = 0^2 + 1 = 1 \]

Setting the Limits Equal
For the function \( f(x) \) to be continuous at \( x = 0 \), the right-hand and left-hand limits must be equal. Hence,

-2a = 1

Solving for \( a \):
Finally, solve for \( a \):

\[ a = - \frac{1}{2} \]

By following these steps, we ensure the function is continuous at \( x = 0 \). Evaluating limits is a powerful technique in ensuring the smooth behavior of functions and solving such problems systematically.