## Precalculus (6th Edition) Blitzer

Yes, the function $f\left( x \right)=3x+4$ is continuous at $1$.
Consider the function $f\left( x \right)=3x+4$, First check whether the function is defined at the point $a$ or not. Find the value of $f\left( x \right)$ at $a=1$, \begin{align} & f\left( 1 \right)=3\left( 1 \right)+4 \\ & =3+4 \\ & =7 \end{align} The function is defined at the point $a=1$. Now find the value of $\,\underset{x\to 1}{\mathop{\lim }}\,3x+4\,$, \begin{align} & \,\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right) \\ & =3\left( 1 \right)+4 \\ & =3+4 \\ & =7 \end{align} Thus, $\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right)\,=7$ Now check whether $\,\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ or not. From the above, $\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right)\,=7\text{ and }f\left( 1 \right)=7$ Therefore, $\,\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=f\left( 1 \right)$ Thus, the function satisfies all the properties of being continuous. Hence, the function $f\left( x \right)=3x+4$ is continuous at $1$.