Is it proper to say "increases/decreases from no bound"?We commonly use the expression increases without bound to describe certain divergent behaviour of functions (e.g. the function f ( x ) = x 2 increases without bound on [ 0 , ∞ )). What would be the proper way of describing the behaviour of the curve of f(x) if I want to move towards the right on ( − ∞ , 0 ]?In the curve sketching unit in my calculus course, I tend to describe all key elements (critical numbers intervals of increase/decrease, points of inflection, intercepts, etc.) from left to right on the x-axis, and so to increase consistency, I want to describe the increase/decrease of a function from left to right. So far, I have been saying "the function increases/decreases without bound towards the left", but this has been directly opposite to what that interval of the function is labelled. I want to know if there is better or more accurate language I could use.

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Is it proper to say &quot;increases/decreases from no bound&quot;?We commonly use the expression increases without bound to describe certain divergent behaviour of functions (e.g. the function   f  (  x  )  =      x    2   increases without bound on   [  0  ,  ∞  )). What would be the proper way of describing the behaviour of the curve of f(x) if I want to move towards the right on   (  −  ∞  ,  0  ]?In the curve sketching unit in my calculus course, I tend to describe all key elements (critical numbers intervals of increase/decrease, points of inflection, intercepts, etc.) from left to right on the x-axis, and so to increase consistency, I want to describe the increase/decrease of a function from left to right. So far, I have been saying &quot;the function increases/decreases without bound towards the left&quot;, but this has been directly opposite to what that interval of the function is labelled. I want to know if there is better or more accurate language I could use.

Gemma Conley · Accepted Answer

Step 1To describe one-sided asymptotes, you can use one-sided limit notation. For example       lim          x      →              a        +              f  (  x  )  =  ∞ is both brief and clear.Or equivalently in words, asxapproaches a from the right, f(x) approaches infinity.Note that for the example above, we are describing right-to-left behavior, but that&#039;s the nature of the behavior.Step 2To describe the behavior as x approaches   ∞ or   −  ∞, there is only one side to approach from, so there&#039;s no need to specify the direction of approach. For example, you might have       lim          x      →      −      ∞        f  (  x  )  =  ∞ or equivalently in words, as x approaches   −  ∞, f(x) approaches   ∞.Note that for this example, once again we are describing right-to-left behavior.

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