Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each limit. (a) \(\lim _{x \rightarrow 9} e^{\sqrt{x}}\) (b) \(\lim _{x \rightarrow-2} e^{\sqrt{x}}\) (c) \(\lim _{x \rightarrow 0} e^{\sqrt{x}}\)

Step-by-Step Solution

Answer

(a) \( e^3 \); (b) Does not exist; (c) 1.

Step 1: Understanding the Limit Process

When evaluating limits, we want to know the value of a function as the variable approaches a certain number. Here, the function is the exponential function raised to the square root of x, written as \(e^{\sqrt{x}}\).

Step 1: Evaluate Limit as x Approaches 9

For \( \lim_{x \rightarrow 9} e^{\sqrt{x}} \), substitute \( x = 9 \) directly since the square root and exponential functions are continuous for positive values. This gives \( e^{\sqrt{9}} = e^3 \).

Step 2: Evaluate Limit as x Approaches -2

For \( \lim_{x \rightarrow -2} e^{\sqrt{x}} \), note that the square root of a negative number is not defined in the real numbers. Thus, this limit does not exist.

Step 3: Evaluate Limit as x Approaches 0

For \( \lim_{x \rightarrow 0} e^{\sqrt{x}} \), as \( x \) approaches 0 from the positive side (since square root is not defined for negative numbers), \( \sqrt{x} \) approaches 0. Therefore, \( e^{\sqrt{x}} \) approaches \( e^0 = 1 \).

Key Concepts

Exponential Functions

Exponential functions are essential in mathematics due to their unique growth patterns. The function is written as \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. Exponential functions have the characteristic that they grow rapidly as the value of \( x \) increases. When we evaluate exponential functions of the form \( e^{\sqrt{x}} \), we are dealing with a composite function, combining both the exponential and square root operations.
To understand this, think of \( e^{\sqrt{x}} \) as two separate steps:

• First, compute the square root of \( x \).
• Then, raise \( e \) to the power of this result.

Both steps are straightforward when \( x \) is a positive number. However, things get tricky with negative \( x \) because the square root function, when combined with \( e \), cannot accept negative inputs when staying within the real numbers.

Square Root Function

The square root function, denoted as \( \sqrt{x} \), is a function that returns the value whose square is \( x \). The square root is only defined for non-negative numbers in the real number system. This means \( \sqrt{x} \) can only be applied to values \( x \geq 0 \) unless we extend our understanding to complex numbers. Such limitations impact calculations, especially in calculus, which often assumes functions over real numbers.
When evaluating limits such as \( \lim _{x \rightarrow c} \sqrt{x} \), it is vital to ensure that \( c \) is a non-negative number to avoid undefined behavior. For instance, taking the square root of 9 is straightforward, \( \sqrt{9} = 3 \), but attempting \( \sqrt{-2} \) leads to complications since this is not a value in the set of real numbers. This constraint leads influenced whether certain limits, particularly those approaching negative numbers, exist.

Evaluating Limits

Evaluating limits in calculus involves determining what value a function approaches as the input gets closer to a specific point. For an expression like \( \lim _{x \rightarrow c} e^{\sqrt{x}} \), you perform the evaluation by substituting values or approaching values that make the function defined and continuous.
Consider these steps:

• Identify the value \( c \) the variable \( x \) is approaching.
• Check the function's behavior near this point. If the function is continuous and defined, substitute \( c \) into the function directly.
• If the function approaches a point from its domain where it becomes discontinuous or undefined, assess the situation: For real numbers, know if the function remains undefined, as seen with \( \lim _{x \rightarrow -2} e^{\sqrt{x}} \).

While some limits can be evaluated directly, others might not exist. For example, the limit of \( e^{\sqrt{x}} \) as \( x \) approaches zero is straightforward since \( e^{0} = 1 \), given that square roots are defined for \( x = 0 \) from the right.