Understanding Limits: Why |h|/h Approaches -1 as h Approaches 0

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In summary, the conversation discusses the value of |h|/h as h approaches 0 from the left. The listener is confused as to why the answer is not -h/-h = 1, as both the numerator and denominator become negative. The expert summarizer clarifies that the numerator is always positive and explains how -x is actually a positive value when x<0. The listener then understands and thanks the expert for their help.
  • #1
ratios
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Hello all,

I don't understand why |h|/h will become -1 as h approches 0 from the left. I can see how the numerator becomes negative, as |x| = -x if x< 0, but shouldn't the denominator become negative also since h < 0 when we approach 0 from the left? So why isn't the answer -h/-h = 1?
 
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  • #2
The numerator is always positive. It's true that |x|=-x when x<0, but in that case -x>0, and |x|/x=-x/x=-1.
 
  • #3
ratios said:
I can see how the numerator becomes negative, as |x| = -x if x< 0
You see wrong. Here, -x is positive.
 
  • #4
Okay I see. Thanks for the help!
 

FAQ: Understanding Limits: Why |h|/h Approaches -1 as h Approaches 0

What is a limit in mathematics?

A limit is a fundamental concept in mathematics that refers to the value that a function or sequence approaches as its input or index approaches a certain value. It is often used to describe the behavior of a function near a specific point.

How do you find the limit of a function?

To find the limit of a function, you can use various methods such as direct substitution, factoring, and L'Hopital's rule. You can also use a graphing calculator or a table of values to estimate the limit.

What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of a function from one direction, either from the left or right side of the specific point, while a two-sided limit takes into account both directions. One-sided limits are denoted by a minus or plus sign next to the point, while two-sided limits are notated with a double-sided arrow.

Can a function have a limit at a point but not be continuous?

Yes, a function can have a limit at a point but not be continuous. This can happen if the function has a hole, a jump, or an asymptote at that point. In order for a function to be continuous at a point, it must have a limit at that point and the function's value must equal the limit.

How do limits relate to derivatives and integrals?

Limits are closely related to derivatives and integrals. Derivatives can be thought of as the instantaneous rate of change at a specific point, which is calculated using a limit. Integrals, on the other hand, are used to find the area under a curve by summing up an infinite number of rectangles with infinitely small widths, which is also based on limits.

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