Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit. \(\lim _{x \rightarrow 2}\left(x^{2}+1\right)\left(x^{2}-4\right)\)

Step-by-Step Solution

Answer

The short answer to the question is: \(\lim _{x \rightarrow 2}\left(x^{2}+1\right)\left(x^{2}-4\right)=0\).

Step 1: Find the limit of the first expression

: We need to find the limit of the first expression, \(x^2 + 1\), as \(x\) approaches 2. To do this, simply substitute \(x = 2\) into the expression: \[\lim _{x \rightarrow 2}\left(x^{2}+1\right) = 2^2 + 1 = 4 + 1 = 5\]

Step 2: Find the limit of the second expression

: Now, we need to find the limit of the second expression, \(x^2 - 4\), as \(x\) approaches 2. This expression is a difference of squares, so we can rewrite it as \((x + 2)(x - 2)\). Then, substitute \(x = 2\) into the expression: \[\lim _{x \rightarrow 2}\left(x^{2}-4\right) = \lim _{x \rightarrow 2}\left((x + 2)(x - 2)\right) = (2 + 2)(2 - 2) = 4 \cdot 0 = 0\]

Step 3: Multiply the two limits

: Finally, to find the limit of the entire function, we multiply the limits of the two expressions that we found in Steps 1 and 2: \[\lim _{x \rightarrow 2}\left(x^{2}+1\right)\left(x^{2}-4\right) = \lim _{x \rightarrow 2}\left(x^{2}+1\right) \cdot \lim _{x \rightarrow 2}\left(x^{2}-4\right) = 5 \cdot 0 = 0\] So the limit of the given function as \(x\) approaches 2 is 0.

Key Concepts

Limit Evaluation

Limits help us understand what a function approaches as the variable gets closer to a certain value. They are foundational in calculus.To evaluate the limit \( \lim_{x \rightarrow 2} (x^2 + 1) \), we simply replace every instance of \( x \) with \(2\).

• First, calculate \(x^2 + 1\) at \(x = 2\)
• This becomes \(2^2 + 1 = 4 + 1 = 5\)

Evaluating limits like this is straightforward when there is no indeterminate form like \(\frac{0}{0}\). If functions are continuous, as is often in simple polynomial cases, directly substituting the variable is a valid method.

Difference of Squares

The difference of squares occurs in expressions like \(x^2 - 4\) and can be factored neatly into \( (x+2)(x-2) \). A difference of squares factors according to the rule: \(a^2 - b^2 = (a+b)(a-b)\).

• This expression becomes zero when either factor is zero.
• We notice that when \(x = 2\), the term \( (x - 2) \) equals zero.

Decomposing expressions through factoring can simplify evaluation, allowing us to handle potential indeterminate forms or make predictions about limits.

Substitution Method

The substitution method is a convenient way to directly compute limits by plugging in the value the variable approaches.It is valid when there is no division by zero or other complication.

• In our example, we evaluate \( x^2 + 1 \) and \( x^2 - 4 \) separately, substituting \( x = 2 \).
• For \( x^2 + 1\), substituting gives \(5\).
• For \( x^2 - 4 \), substituting gives zero due to the difference of squares calculation.

Once both parts are evaluated, you can multiply the results to find the overall limit. This method is especially powerful when expressions can be easily simplified by direct substitution.