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Understanding the limit of a function is a core concept in calculus, which in simple terms helps us determine what output values a function approaches as the input values approach a certain point. For example, when we see notation such as \(\lim _{x \rightarrow a} f(x)\), it asks: what value does the function \(f(x)\) approach as \(x\) gets closer and closer to \(a\)? This becomes particularly useful when dealing with continuous and discontinuous points.

In our exercise, for part (a), we have the limit \(\lim _{x \rightarrow 1} f(x)\) of a simplified rational function. Here, we can directly substitute \(x = 1\) into the simplified function \(f(x) = x - 4\) because the function is continuous at \(x = 1\). As a result, we easily find that the limit is \(f(1) = -3\). This direct substitution method is often the easiest way to calculate limits when the function is well-behaved and continuous at the point of interest.

Rational functions are ratios of polynomials, and simplification is essential for analyzing their limits and overall behavior. Simplifying a rational function involves factoring both the numerator and the denominator and then canceling out any common factors.

Consider the function given in our exercise, \(f(x) = \frac{x^2 - 3x - 4}{x + 1}\). The numerator \(x^2 - 3x - 4\) factors into \(x-4)(x+1)\), revealing a common factor of \(x+1\) in the numerator and denominator. Upon simplification, we end up with the function \(f(x) = x - 4\), excluding \(x = -1\) from its domain as it would make the original denominator zero. Simplifying rational functions before calculating limits can often lead to straightforward solutions and is a valuable algebraic skill in calculus.

Limits can become undefined when approaching values that would result in a division by zero or some other indeterminate form. In calculus, an undefined limit means there is no finite limit at the point under consideration. It is essential to investigate the original function before simplification to determine if any values of \(x\) were excluded due to division by zero.

In part (b) of our exercise, \(\lim _{x \rightarrow -1} f(x)\) was investigated after simplifying the function. However, since the factor \(x+1\) was canceled during simplification, it must be noted that the function cannot be evaluated at \(x = -1\) in its original form, as it would involve division by zero. Thus, the limit as \(x\) approaches -1 is undefined. This underscores the importance of considering the domain of the original function when working with limits and symbolizes the delicate nature of handling undefined limits in calculus.