Find each limit. Use $-\infty$ or $\infty$ when appropriate.
$f(x)=-\frac{2 x-2}{(x-2)^4}$
(A) $\lim_{x \rightarrow 2^{-}} f(x)$
(B) $\lim_{x \rightarrow 2^{+}} f(x)$
(C) $\lim_{x \rightarrow 2} f(x)$
Answer (1)
Analyze the function f ( x ) = − ( x − 2 ) 4 2 x − 2 as x approaches 2.
Determine that as x → 2 + , ( x − 2 ) 4 approaches 0 from the positive side, while − ( 2 x − 2 ) approaches -2.
Conclude that lim x → 2 + f ( x ) = − ∞ .
Since both one-sided limits are − ∞ , then lim x → 2 f ( x ) = − ∞ .
Explanation
Problem Analysis We are given the function f ( x ) = − ( x − 2 ) 4 2 x − 2 and asked to find the limits as x approaches 2 from the left, from the right, and the overall limit as x approaches 2. We are given that lim x → 2 − f ( x ) = − ∞ .
Limit from the Right To find lim x → 2 + f ( x ) , we need to analyze the behavior of the function as x approaches 2 from the right. As x approaches 2 from the right, x − 2 approaches 0 through positive values. Thus, ( x − 2 ) 4 also approaches 0 through positive values. The numerator 2 x − 2 approaches 2 ( 2 ) − 2 = 2 , which is positive. Therefore, the expression − ( 2 x − 2 ) approaches − 2 , which is negative. So we have a negative number divided by a very small positive number, which means the limit is − ∞ .
Calculating the Right-Hand Limit x → 2 + lim f ( x ) = x → 2 + lim − ( x − 2 ) 4 2 x − 2 = − ∞
Overall Limit To find lim x → 2 f ( x ) , we need to compare the left-hand limit and the right-hand limit. We are given that lim x → 2 − f ( x ) = − ∞ , and we found that lim x → 2 + f ( x ) = − ∞ . Since both the left-hand and right-hand limits are equal to − ∞ , the overall limit is also − ∞ .
Calculating the Overall Limit x → 2 lim f ( x ) = − ∞
Final Answer Therefore, the answers are: (A) lim x → 2 − f ( x ) = − ∞ (B) lim x → 2 + f ( x ) = − ∞ (C) lim x → 2 f ( x ) = − ∞
Examples
Understanding limits is crucial in physics, especially when dealing with asymptotic behaviors. For instance, when analyzing the motion of an object approaching the speed of light, certain physical quantities may approach infinity. Similarly, in circuit analysis, the current through a capacitor as it charges can be modeled using limits to understand its behavior over time. These concepts are not just theoretical; they are fundamental to predicting and understanding real-world phenomena.
