Problem 40 - Calculus for Biology and Medicine

Question

In Problems \(29-48\), find the limits. $$ \lim _{x \rightarrow-1} e^{x^{2} / 2-1} $$

Step-by-Step Solution

Answer

The limit is \( e^{-1/2} \).

Step 1: Apply Limit Inside the Exponent

To find the limit as \( x \to -1 \) of \( e^{x^2/2 - 1} \), we start by evaluating the limit of the exponent expression \( \frac{x^2}{2} - 1 \).

Step 2: Substitute the Limit into the Exponent

Substitute \( x = -1 \) into the expression to evaluate the exponent: \( \frac{(-1)^2}{2} - 1 = \frac{1}{2} - 1 = -\frac{1}{2} \).

Step 3: Evaluate the Exponential Function

Now substitute the evaluated limit of the exponent back into the exponential function: \( e^{x^2/2 - 1} = e^{-1/2} \).

Step 4: Final Result

Therefore, the limit \( \lim_{x \to -1} e^{x^2/2 - 1} = e^{-1/2} \).

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in calculus and mathematics in general. These functions involve the expression \( a^x \), where \( a \) is a constant base and \( x \) is the exponent. The most notable base is Euler's number, \( e \), approximately equal to 2.718. Exponential functions with this base are called natural exponential functions, written as \( e^x \), and they have unique properties, such as their rate of growth being proportional to their current value. You will often see exponential functions in scenarios involving growth and decay, such as population growth or radioactive decay.

In calculus, exponential functions are intriguing because their derivatives and integrals are also exponential functions. This makes them relatively simple to handle, as their forms don't change much through differentiation or integration.

Limit Evaluation

Evaluating limits is a foundational topic in calculus, which involves determining the value that a function approaches as the input approaches a certain point. Limits help us understand the behavior of functions at points where they may not be explicitly defined. In our original problem, \( \lim_{x \to -1} e^{x^2/2 - 1} \), the goal is to find the limit of the function as \( x \) approaches -1.

To evaluate this, we first look at the exponent \( x^2/2 - 1 \). By substituting \(-1\) into the expression, it simplifies the process. At \( x = -1 \), the exponential expression becomes \( e^{(-1)^2/2 - 1} = e^{-1/2} \). Thus, calculating the limit involves resolving it step by step, starting with analyzing and simplifying the internal expressions before applying exponential functions.

Substitution Method

In calculus, the substitution method is a powerful technique used often to simplify expressions and evaluate limits. With substitution, you replace variables in an expression with other expressions that make the problem easier to solve. In the given exercise, \( \lim_{x \to -1} e^{x^2/2 - 1} \), the substitution method is applied to determine the limit of the exponent.

Here's how it works step-by-step:

• Identify the inner expression within the function, which in this case is \( x^2/2 - 1 \).
• Substitute the limit point \( x = -1 \) directly into this expression.
• Evaluate the expression to find the new simplified term; here it becomes \(-1/2\).
• Substitute this simplified result back into the original exponential function to find the final expression, \( e^{-1/2} \).

This method reduces complex functions into manageable parts, often streamlining the evaluation process while maintaining mathematical accuracy.