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 \( \frac{1}{\sqrt{e}} \).

Step 1: Substitute x with -1

To find the limit as \( x \) approaches \(-1\), we first substitute \( x = -1 \) directly into the expression. This yields:\[ e^{((-1)^{2} / 2) - 1} \] Simplifying the exponent leads to:\[ ((-1)^{2} = 1) ightarrow rac{1}{2} - 1 \] Thus, the exponent becomes:\[ -\frac{1}{2} \]

Step 2: Simplify the Exponential Expression

Now that we have simplified the exponent to \(-\frac{1}{2}\), we calculate:\[ e^{-\frac{1}{2}} \] This value can be interpreted as the reciprocal of \(e\) raised to the power \(\frac{1}{2}\), which is also equivalent to:\[ \frac{1}{\sqrt{e}} \]

Step 3: Evaluate the Limit

Since the function is continuous and we have substituted the limit point directly into the expression, the limit of the function as \( x \rightarrow -1 \) is simply the value of the expression evaluated at \( x = -1 \):\[ \frac{1}{\sqrt{e}} \]

Key Concepts

Understanding Exponential Functions

Exponential functions are incredibly important in mathematics and appear frequently in problems related to limits and calculus. At their core, these functions involve expressions in which a constant base, like the famous number \( e \), is raised to a variable exponent. The natural exponential, \( e^x \), is particularly significant because \( e \) is an irrational number approximately equal to 2.71828. It has unique properties that make it incredibly useful for solving problems involving growth, decay, and in this case, limits.

• Exponential functions grow extremely quickly or decay rapidly depending on whether the exponent is positive or negative.
• They never touch the x-axis, meaning their values never actually become zero.
• Such functions can effectively model real-world processes like population growth or radioactive decay, thanks to their dynamic nature.

In our exercise, we are dealing with a function where the expression \( e^{x^2/2 -1} \) is evaluated as \( x \) approaches -1. This involves simplifying the exponent to determine how the function behaves as the input gets close to this particular value. The behavior of exponential functions under change is essential in finding this limit.

Exploring Continuity in Functions

Continuity in functions is a fundamental concept in calculus, crucial to understanding limits. A function can be described as continuous at a point if there is no interruption or jump at that point. Specifically, a function \( f(x) \) is continuous at a point \( x = a \) if the limit of \( f(x) \) as \( x \) approaches \( a \) is equal to the function's value at that point, i.e., \( \lim_{x \to a} f(x) = f(a) \).

• This means that you can draw the graph of the function without lifting your pencil off the paper.
• Continuity ensures that small changes in the input lead to small changes in the output.

In our example, the function \( e^{x^2/2 -1} \) is continuous at \( x = -1 \). This characteristic allows us to substitute \( x = -1 \) directly into the expression to find the limit. By confirming continuity first, we know that the simplified expression will not produce a jump or undefined behavior at this input point.

The Substitution Method in Limits

The substitution method in limits is a straightforward yet powerful tool for solving limit problems that involve continuous functions. When a function is continuous at the point you are considering, you can directly substitute the limiting value into the function to compute the limit.

• This method saves time and eliminates the need for more complex manipulations or considerations.
• You must be certain that the function does not have breaks, holes, or undefined points at the value where you substitute.

In this exercise, we took advantage of this method. By substituting \( x = -1 \) directly into the function \( e^{x^2/2 -1} \), the problem simplifies to calculating the expression for the exponent at this point. This resulted in \( -\frac{1}{2} \). As the function is continuous at \( x = -1 \), the limit is simply the value of the expression at that point: \( \frac{1}{\sqrt{e}} \). This method emphasizes how understanding the behavior of a function at specific points can allow simple substitution to yield correct and immediate results.