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The phrase "approaching a threshold" is correct and usable in written English.
It can be used when discussing a situation where something is nearing a limit or a significant point of change.
Example: "The company is approaching a threshold in its growth, indicating that it may soon need to expand its operations."
Alternatives: "nearing a limit" or "coming close to a boundary."
Exact(6)
Under weak OH circumstances, the impedance of mass transfer quickly multiplies and the catalytic current declines to a current plateau in the region of 1.7 2.1 V vs. RHE, due to the surface H concentration approaching a threshold value in the range of 10−3 10−4 M.
He was approaching a threshold that renders him quite hittable.
The search for generic, universal indicators that a system is approaching a threshold is an active topic of both theoretical and empirical research.
In sum, research on ecological resilience tends to center on analysis of a natural or ecological system's alternative stable states, the drivers of transitions across thresholds between states, and the characteristics of system structure and dynamics when approaching a threshold.
The dynamical equivalence between models of this sort, where populations experience exponential growth asymptotically approaching a threshold, and those models used in neural networks of excitation/inhibition between neurons is well recognised ([ 64] and refs. within).
As the particles were found to remain insoluble and stable for several months in aqueous solutions (data not shown) this might be the result of approaching a threshold level for the effect of the soluble alginate concentration.
Similar(54)
It is shown that joint stiffness depends non-linearly on the ratio of adhesive to adherend modulus, and approaches a threshold value with increasing adhesive modulus.
Kathleen Newland, also of MPI, suggests that "based on the experience of countries like Spain, Portugal, Greece and South Korea, emigration usually slows when income per person approaches a threshold level in relation to income in the richer countries where the migrants are heading".
Share simulation indicates that railroad share will sharply increase when time reduces and approaches a threshold; furthermore, only when the reduction of railroad transportation chain time exceeds 35%, reducing cost or time fluctuation will significantly increase its share.
"If the shortage of funds persists," promises Russia's Defense Minister, Igor Rodionov, "Russia may soon approach a threshold beyond which its missiles and nuclear systems become uncontrollable".
If one observes that (mathbf{u}_{beta}) approaches a threshold advanced limit, as β increases, or that the entire sequence does not converge, the alarm bell should ring.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com