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Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mo>=</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mi>n</mi> </msub> </semantics> </math> </inline-formula> be a sequence of continuous self-maps on <i>I</i> which converge uniformly to a self-map <i>f</i> on <i>I</i>. Denote by <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">F</mi> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> the set of fuzzy numbers on <i>I</i>, and denote by <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">F</mi> <mrow> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> <mo>,</mo> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">F</mi> <mrow> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> the Zadeh<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>′</mo> </msup> </semantics> </math> </inline-formula>s extensions of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mo>,</mo> <mi>f</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mo>,</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, respectively. In this paper, we study the <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula>-limit sets of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">F</mi> <mrow> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and show that, if all periodic points of <i>f</i> are fixed points, then <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ω</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>,</mo> <msub> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>⊂</mo> <mi>F</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for any <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">F</mi> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ω</mi> <mo>(</mo> <mi>A</mi> <mo>,</mo> <msub> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is the <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula>-limit set of <i>A</i> under <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">F</mi> <mrow> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mi>n</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">F</mi> <mrow> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> <mo>:</mo> <mover accent="true"> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>A</mi> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>.