Q. 3 - Calculus

Question

State the constant multiple rule, sum rule, product rule, quotient rule, and composition rule for limits.

Step-by-Step Solution

Answer

The constant multiple rule is $\underset{x\to c}{\lim }kf\left(x\right)=k\underset{x\to c}{\lim }f\left(x\right)$.

The sum rule is $\underset{x\to c}{\lim }\left[f\right(x)+g(x\left)\right]=\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)$.

The product rule is $\underset{x\to c}{\lim }\left[f\right(x\left)g\right(x\left)\right]=\underset{x\to c}{\lim }f\left(x\right)\underset{x\to c}{\lim }f\left(x\right)$.

The quotient rule is $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to c}{\lim }f\left(x\right)}{\underset{x\to c}{\lim }g\left(x\right)};g\left(x\right)\ne 0$.

The composition rule for limits is $\underset{x\to c}{\lim }f\left[g\right(x\left)\right]=f\left(\underset{x\to c}{\lim }g\right(x\left)\right)$, where f is continuous at $\underset{x\to c}{\lim }g\left(x\right)$.

Step 1. Given information.

We have to write sum rule, constant rule, product rule, quotient rule, and composition rule for limits.

Step 2. Statements.

The constant multiple rule is $\underset{x\to c}{\lim }kf\left(x\right)=k\underset{x\to c}{\lim }f\left(x\right)$.

The sum rule is $\underset{x\to c}{\lim }\left[f\right(x)+g(x\left)\right]=\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)$.

The product rule is $\underset{x\to c}{\lim }\left[f\right(x\left)g\right(x\left)\right]=\underset{x\to c}{\lim }f\left(x\right)\underset{x\to c}{\lim }f\left(x\right)$.

The quotient rule is $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to c}{\lim }f\left(x\right)}{\underset{x\to c}{\lim }g\left(x\right)};g\left(x\right)\ne 0$.

The composition rule for limits is $\underset{x\to c}{\lim }f\left[g\right(x\left)\right]=f\left(\underset{x\to c}{\lim }g\right(x\left)\right)$, where f is continuous at $\underset{x\to c}{\lim }g\left(x\right)$.