Suppose f(x)𝑓(𝑥) and g(x)𝑔(𝑥) are equal for all x-values except x=t𝑥=𝑡. If limx→tf(x)=Llim𝑥→𝑡𝑓(𝑥)=𝐿, then is limx→tg(x)=Llim𝑥→𝑡𝑔(𝑥)=𝐿 true?
Question
Suppose f(x)𝑓(𝑥) and g(x)𝑔(𝑥) are equal for all x-values except x=t𝑥=𝑡. If limx→tf(x)=Llim𝑥→𝑡𝑓(𝑥)=𝐿, then is limx→tg(x)=Llim𝑥→𝑡𝑔(𝑥)=𝐿 true?
Solution
Yes, it is true. If f(x) and g(x) are equal for all x-values except x=t, then the limits of f(x) and g(x) as x approaches t will also be equal. This is because the limit of a function at a certain point is determined by the values around that point, not the value at the point itself. So even though f(t) and g(t) might not be equal, lim(x→t) f(x) will still equal lim(x→t) g(x) if f(x) and g(x) are equal for all other x-values.
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