Question

Suppose lim⁡x→a f(x)=L{\displaystyle\lim_{x	o a}}	ext{ }f\left(xight)=L and lim⁡x→a g(x)=M{\displaystyle\lim_{x	o a}}	ext{ }g\left(xight)=M. Prove that lim⁡x→a (f(x)−g(x))=L−M{\displaystyle\lim_{x	o a}}	ext{ }\left(f\left(xight)-g\left(xight)ight)=L-M.

Accepted Answer

Step 1: Start by recalling the limit properties. One important property is that the limit of a difference is the difference of the limits, provided the individual limits exist. This can be expressed as: if $$ \lim_{x 	o a} f(x) = L $$ and $$ \lim_{x 	o a} g(x) = M $$, then $$ \lim_{x 	o a} (f(x) - g(x)) = \lim_{x 	o a} f(x) - \lim_{x 	o a} g(x) . $$Step 2: Apply the limit property to the given functions $$ f(x) $$ and $$ g(x) . $$According to the problem, $$ \lim_{x 	o a} f(x) = L $$ and $$ \lim_{x 	o a} g(x) = M . $$Therefore, using the property, we have $$ \lim_{x 	o a} (f(x) - g(x)) = L - M . $$Step 3: Verify the conditions for applying the limit property. Ensure that both $$ \lim_{x 	o a} f(x) $$ and $$ \lim_{x 	o a} g(x) $$ exist and are finite, which is given in the problem statement. Step 4: Conclude that since the conditions are satisfied and the property is applicable, the limit of the difference $$ \lim_{x 	o a} (f(x) - g(x))  is $$indeed $$ L - M . $$Step 5: Summarize the proof by stating that the limit of the difference of two functions is equal to the difference of their limits, as demonstrated by applying the limit property to the given functions.

Suppose $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ . Prove that $\lim_{x \to a} (f (x) - g (x)) = L - M$ .

Verified video answer for a similar problem:

Key Concepts

Limit of a Function

Limit Properties

Proof Techniques in Calculus