Question

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=4-x^{2}, g(x)=\sqrt{x+1} $$ $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 1} f(x)} & {\text { (b) } \lim _{x \rightarrow 3} g(x) \quad(\mathrm{c}) \lim _{x \rightarrow 1} g(f(x))}\end{array} $$

Short answer

The answers are \(\lim_{x \rightarrow 1} f(x) = 3\), \(\lim_{x \rightarrow 3} g(x) = 2\), and \(\lim_{x \rightarrow 1} g(f(x)) = 2\).

Step-by-step solution

Evaluate \(\lim_{x \rightarrow 1} f(x)\)

To find the limit of a function at a certain point, one usually substitutes the point into the function. Therefore, substitute \(x = 1\) into \(f(x) = 4 - x^2\) to get \(f(1) = 4 - 1^2 = 3\). Thus, \(\lim_{x \rightarrow 1} f(x) = 3.\)

Evaluate \(\lim_{x \rightarrow 3} g(x)\)

Similarly, substitute \(x = 3\) into \(g(x) = \sqrt{x+1}\) to get \(g(3) = \sqrt{3 + 1} = \sqrt{4} = 2\). Thus, \(\lim_{x \rightarrow 3} g(x) = 2.\)

Evaluate \(\lim_{x \rightarrow 1} g(f(x))\)

This is a composite function, \(g(f(x))\), so we first substitute \(x = 1\) into \(f(x)\) to get \(f(1) = 3\) (which we found in Step 1). We then substitute this result, \(f(1) = 3\), into \(g(x)\) to get \(g(3) = 2\) (which we found in Step 2). Thus, \(\lim_{x \rightarrow 1} g(f(x)) = 2.\)

Key concepts

Limit of a Function

The concept of a limit is foundational in calculus and is crucial for understanding the behavior of functions as the inputs approach a particular value. A limit tries to find out what the function value is moving towards as the variable gets arbitrarily close to a given number.

When we say \( \lim_{x \rightarrow c} f(x) = L \), we're stating that as \(x\) approaches \(c\), \(f(x)\) approaches the value \(L\). However, it's important to note that \(f(x)\) does not necessarily have to equal \(L\) when \(x = c\); what matters is the trend as \(x\) gets close to \(c\).

For evaluating basic limits, if the function is continuous at the point to which \(x\) is approaching, you can simply substitute the value of \(x\) into the function to find the limit, as was done in the step-by-step solution for the exercise. However, if the function isn't continuous or has a more complex structure, other techniques like factoring, conjugate multiplication, or advanced theorems may be necessary to find the limit.

Substitution Method

The substitution method is often the first and simplest technique employed when finding the limit of a function. It involves directly replacing the variable in the function with the value it is approaching, assuming that direct substitution doesn't lead to an undefined expression or an indeterminate form such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

In the provided textbook exercise, direct substitution works neatly because both functions \(f(x)\) and \(g(x)\) are well-defined and continuous at the points of interest, meaning there are no holes, jumps, or vertical asymptotes that would prevent direct substitution. By simply plugging \(x = 1\) and \(x = 3\) into the respective functions, you quickly find the limits to be 3 and 2, respectively. It is always a good practice to verify the continuity of the function at that point before applying this method.

Composite Functions

A composite function can be thought of as a function within a function, written as \(g(f(x))\), which means you apply the function \(f\) to \(x\) and then apply the function \(g\) to the result of \(f(x)\). When dealing with limits involving composite functions, you often work from the inside out.

As illustrated in the exercise, to evaluate the limit of a composite function at a point, you first evaluate the inner function at that point, as was done with \(f(x)\) when \(x\) was approaching 1. With the output from \(f(x)\), you can then evaluate the outer function \(g(x)\).

This process might sometimes require back-and-forth substitution if the inner function itself has significant complexity. Nonetheless, in the given problem, we have a relatively straightforward scenario where finding the limit of the composite function \(\lim_{x \rightarrow 1} g(f(x))\) was successfully achieved by sequentially applying the functions \(f\) and \(g\).