Problem 32 - Single Variable Calculus

Question

Find the limit. $$ \lim _{t \rightarrow-\infty} \frac{2 t^{2}}{\sqrt{t^{4}+t^{2}}} $$

Step-by-Step Solution

Answer

\( \lim_{t \rightarrow -\infty} \frac{2t}{-\sqrt{t^2 + 1}} \) does not exist.

Step 1: Simplify the function

First, we want to simplify the given function. We can take the greatest common power of t from the numerator and denominator to achieve this. Given function: \(\frac{2t^2}{\sqrt{t^4 + t^2}}\) Take \(t^2\) as a common factor: \[ \frac{2t^2}{\sqrt{t^2(t^2 + 1)}} \] Now, we can simplify the expression further by taking \(t^2\) out of the square root: \[ \frac{2t^2}{|t|\sqrt{t^2 + 1}} \]

Step 2: Handle the absolute value sign

Since we are finding the limit as \(t\) approaches -∞, we can replace the absolute value sign: \(|t| = -t\) since \(t < 0\) Thus, the expression becomes: \[ \frac{2t^2}{-t\sqrt{t^2 + 1}} \] Now, let's cancel out the t-term in the expression: \[ \frac{2t}{-\sqrt{t^2 + 1}} \]

Step 3: Evaluate the limit

Now we can evaluate the limit as \(t\) approaches -∞: \[ \lim_{t \rightarrow -\infty} \frac{2t}{-\sqrt{t^2 + 1}} \] To find this limit, we can divide both the numerator and the denominator by \(t\): \[ \lim_{t \rightarrow -\infty} \frac{2}{-\frac{1}{t}\sqrt{t^2 + 1}} \] Now, as \(t\) approaches -∞, \(-\frac{1}{t}\) approaches 0. Therefore, the expression simplifies to: \[ \lim_{t \rightarrow -\infty} \frac{2}{-\frac{1}{t}\sqrt{t^2 + 1}} = \frac{2}{0} \] Since we cannot have a denominator of 0, the limit does not exist. Therefore, the answer is: The limit does not exist.

Key Concepts

Evaluating Limits

Evaluating limits is a foundational concept in calculus, where we determine what value a function approaches as the input approaches a certain point. This technique is especially used when dealing with functions that may not be explicitly defined at that point. It requires analyzing the behavior of the function and can involve algebraic manipulations to simplify the function to a form where direct substitution can be applied. Often, when direct substitution doesn't work, we use techniques like factoring, multiplying by a conjugate, or simplifying complex fractions. Evaluating limits is crucial for understanding the continuity and behavior of functions near particular points.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the variable approaches a certain point or infinite. In mathematical context, we often look for how the outputs of a function behave as the input becomes very large or very small. When dealing with infinite limits, you usually see terms like

• \(\lim_{t \rightarrow \infty} f(t) = \infty\)
• \(\lim_{t \rightarrow -\infty} f(t) = -\infty\)

This indicates that the function values go towards positive or negative infinity, respectively. To find these limits, understanding the dominant terms in expressions, particularly polynomials and rational functions, is essential. This concept helps us analyze functions at the edges of their domain.

Absolute Value in Limits

Absolute value transforms numbers to their non-negative equivalents, which plays a vital role in certain limits. When calculating limits approaching negative infinity for expressions involving absolute values, we need special consideration. For example,

• For \(|t|\), the expression equals \(-t\) if \(t\) is negative.

This adjustment ensures the expression remains true to its absolute value definition. Understanding how absolute value affects limits helps us simplify expressions correctly, especially when dealing with exponents or polynomials, ensuring accurate simplification and evaluation.

Rational Expressions in Limits

Rational expressions are fractions with polynomials in their numerator and denominator. Evaluating limits involving rational expressions requires simplifying these expressions by factoring or dividing terms appropriately. In many cases, it's beneficial to divide every term by the highest power of the variable to simplify the limit, especially when approaching infinity. In our example, dividing by \(t\) helped identify the function's behavior at infinity by reducing complex terms. Being adept with rational expressions assists in analyzing the dominant factors, crucial for evaluating limits both at finite points and infinity.