Let $f(x)=\left\{\begin{array}{lll}7 x-6 & \text { if } & x<8 \\ \frac{3}{x+9} & \text { if } & x \geq 8\end{array}\right.$
Show that $f(x)$ has a jump discontinuity at $x=8$ by calculating the limits from the left and right at $x=8$.
$\lim _{x \rightarrow 8^{-}} f(x)=$
$\square$
$\lim _{x \rightarrow 8^{+}} f(x)=$
$\square$
Answer (2)
The left-hand limit of the function f ( x ) as x approaches 8 is 50, while the right-hand limit is 17 3 . Since these two limits are not equal, the function has a jump discontinuity at x = 8 . Hence, the limits are: lim x → 8 − f ( x ) = 50 and lim x → 8 + f ( x ) = 17 3 .
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Calculate the left-hand limit as x approaches 8: lim x → 8 − f ( x ) = lim x → 8 − ( 7 x − 6 ) = 7 ( 8 ) − 6 = 50 .
Calculate the right-hand limit as x approaches 8: lim x → 8 + f ( x ) = lim x → 8 + x + 9 3 = 8 + 9 3 = 17 3 .
Observe that the left-hand limit and the right-hand limit are not equal: 50 = 17 3 .
Conclude that f ( x ) has a jump discontinuity at x = 8 , with lim x → 8 − f ( x ) = 50 and lim x → 8 + f ( x ) = 17 3 .
Explanation
Problem Analysis We are given a piecewise function f ( x ) and we want to show that it has a jump discontinuity at x = 8 . To do this, we need to calculate the left-hand limit and the right-hand limit at x = 8 and show that they are not equal.
Calculating the Left-Hand Limit The left-hand limit is the limit of f ( x ) as x approaches 8 from the left, i.e., x < 8 . In this case, f ( x ) = 7 x − 6 . So, we have x → 8 − lim f ( x ) = x → 8 − lim ( 7 x − 6 ) Since 7 x − 6 is a continuous function, we can evaluate the limit by direct substitution: x → 8 − lim ( 7 x − 6 ) = 7 ( 8 ) − 6 = 56 − 6 = 50 Thus, the left-hand limit is 50.
Calculating the Right-Hand Limit The right-hand limit is the limit of f ( x ) as x approaches 8 from the right, i.e., x ≥ 8 . In this case, f ( x ) = x + 9 3 . So, we have x → 8 + lim f ( x ) = x → 8 + lim x + 9 3 Since x + 9 3 is a continuous function at x = 8 , we can evaluate the limit by direct substitution: x → 8 + lim x + 9 3 = 8 + 9 3 = 17 3 ≈ 0.1765 Thus, the right-hand limit is 17 3 .
Conclusion Since the left-hand limit is 50 and the right-hand limit is 17 3 , and 50 = 17 3 , the function f ( x ) has a jump discontinuity at x = 8 .
Final Answer Therefore, x → 8 − lim f ( x ) = 50 x → 8 + lim f ( x ) = 17 3
Examples
Jump discontinuities are important in many engineering applications, such as signal processing. For example, the voltage in an electrical circuit may change abruptly at a certain time, creating a jump discontinuity. Understanding how to analyze these discontinuities is crucial for designing and analyzing such circuits.
