Question 1 State if the statement below is true, or false. If it is false, write the correct statement. \( 1.1 \lim _{x \rightarrow a} f(x)=f(a) \). \( 1.2 \lim _{x \rightarrow a}(f(x)-g(x))=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x) \). \( 1.3 \lim _{x \rightarrow c}\left(\frac{f(x)}{g(x)}\right)=\frac{\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)}{\lim _{x \rightarrow a} g(x)} \). \( 1.4 \lim _{x \rightarrow a} f(x) \cdot g(x)=\left(\lim _{x \rightarrow a} f(x)\right)\left(\lim _{x \rightarrow a} g(x)\right) \). \( 1.5 \lim _{x \rightarrow a} f(x)^{n}=(n-1) \lim _{x \rightarrow a} f(x)^{(n-1)} \.

We will analyze each statement one by one. 1.1. Statement:   limₓ→ₐ f(x) = f(a) This statement is not universally true. It holds only if f is continuous at x = a.   Correct statement: If f is continuous at a, then limₓ→ₐ f(x) = f(a). 1.2. Statement:   limₓ→ₐ (f(x) – g(x)) = limₓ→ₐ f(x) + limₓ→ₐ g(x) This statement is false because the correct limit law for differences involves a subtraction, not addition.   Correct statement: Provided the limits exist, limₓ→ₐ (f(x) – g(x)) = limₓ→ₐ f(x) – limₓ→ₐ g(x). 1.3. Statement:   limₓ→c (f(x)/g(x)) = (limₓ→ₐ f(x) – limₓ→ₐ g(x)) / (limₓ→ₐ g(x)) There are two issues here. First, the limit on the left is taken as x → c, while on the right the limits are written as x → a. Second, the operation in the numerator is subtraction, but for quotients we should be dividing the individual limits, not subtracting them.   Correct statement: If limₓ→c f(x) and limₓ→c g(x) exist and limₓ→c g(x) ≠ 0, then     limₓ→c (f(x)/g(x)) = (limₓ→c f(x)) / (limₓ→c g(x)). 1.4. Statement:   limₓ→ₐ (f(x) · g(x)) = (limₓ→ₐ f(x)) · (limₓ→ₐ g(x)) This statement is true provided the individual limits exist. 1.5. Statement:   limₓ→ₐ (f(x))ⁿ = (n – 1) · limₓ→ₐ (f(x))^(n–1) This statement is false. The correct application of the power rule for limits is:   Correct statement: If the limit limₓ→ₐ f(x) exists, then     limₓ→ₐ (f(x))ⁿ = (limₓ→ₐ f(x))ⁿ. In summary:   1.1 False – Correct if f is continuous at a.   1.2 False – Corrected to subtraction: limₓ→ₐ (f(x) – g(x)) = limₓ→ₐ f(x) – limₓ→ₐ g(x).   1.3 False – Corrected to: limₓ→c (f(x)/g(x)) = (limₓ→c f(x))/(limₓ→c g(x)), provided the limit of g(x) is nonzero.   1.4 True.   1.5 False – Corrected to: limₓ→ₐ (f(x))ⁿ = (limₓ→ₐ f(x))ⁿ.