Let $f(x)=\left\{\begin{array}{lll}4 x-2 & \text { if } & x<4 \\ \frac{1}{x+4} & \text { if } & x \geq 4\end{array}\right.
Show that $f(x)$ has a jump discontinuity at $x=4$ by calculating the limits from the left and right at $x=4$.
$\begin{array}{l}
\lim _{x \rightarrow 4^{-}} f(x)= \\
\lim _{x \rightarrow 4^{+}} f(x)=
\end{array}$
Now for fun, try to graph $f(x)$.
Answer (2)
The left-hand limit as x approaches 4 is 14, and the right-hand limit is 8 1 . Since these two limits are not equal, the function f ( x ) has a jump discontinuity at x = 4 .
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Calculate the left-hand limit as x approaches 4: lim x → 4 − f ( x ) = lim x → 4 − ( 4 x − 2 ) = 14 .
Calculate the right-hand limit as x approaches 4: lim x → 4 + f ( x ) = lim x → 4 + x + 4 1 = 8 1 .
Observe that the left-hand limit and the right-hand limit are not equal: 14 = 8 1 .
Conclude that f ( x ) has a jump discontinuity at x = 4 : Jump discontinuity at x = 4 .
Explanation
Problem Analysis We are given a piecewise function f ( x ) and we want to show that it has a jump discontinuity at x = 4 . To do this, we need to calculate the left-hand limit and the right-hand limit at x = 4 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 4 from the left, i.e., x < 4 . In this case, f ( x ) = 4 x − 2 . Therefore, we have x → 4 − lim f ( x ) = x → 4 − lim ( 4 x − 2 ) Since 4 x − 2 is a continuous function, we can evaluate the limit by direct substitution: x → 4 − lim ( 4 x − 2 ) = 4 ( 4 ) − 2 = 16 − 2 = 14 So, the left-hand limit is 14.
Calculating the Right-Hand Limit The right-hand limit is the limit of f ( x ) as x approaches 4 from the right, i.e., 4"> x > 4 . In this case, f ( x ) = x + 4 1 . Therefore, we have x → 4 + lim f ( x ) = x → 4 + lim x + 4 1 Since x + 4 1 is a continuous function for -4"> x > − 4 , we can evaluate the limit by direct substitution: x → 4 + lim x + 4 1 = 4 + 4 1 = 8 1 = 0.125 So, the right-hand limit is 8 1 .
Conclusion Since the left-hand limit is 14 and the right-hand limit is 8 1 , and 14 = 8 1 , the function f ( x ) has a jump discontinuity at x = 4 .
Final Answer The left-hand limit is x → 4 − lim f ( x ) = 14 The right-hand limit is x → 4 + lim f ( x ) = 8 1 Since the left and right limits are different, there is a jump discontinuity at x = 4 .
Examples
Jump discontinuities are important in real-world applications such as signal processing, where signals can have abrupt changes. For example, the voltage in an electrical circuit might change suddenly due to a switch being flipped. Understanding jump discontinuities helps engineers analyze and design systems that can handle such abrupt changes effectively. Another example is in computer graphics, where jump discontinuities can represent sharp edges or boundaries between different objects in an image.
