Question

# Use the definition of continuity and the properties of limits to show that the function is continuous at the given number. f(x)=x^{2}+\sqrt{7-x}, a=4

Composite functions
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}+\sqrt{{{7}-{x}}},{a}={4}$$

$$\displaystyle\lim_{{{x}\rightarrow{4}}}{x}^{{{2}}}+\sqrt{{\lim_{{{x}\rightarrow{4}}}{7}-\lim_{{{x}\rightarrow{4}}}{x}}}$$ Apply the Sum Law, Difference Law, and Root Law
$$\displaystyle{\left({4}\right)}^{{{2}}}+\sqrt{{{7}-{4}}}$$ Plug in the corresponding values
$$\displaystyle{f{{\left({4}\right)}}}={16}+\sqrt{{{3}}}$$ Evaluate
continuous f(4) is defined and equal to $$\displaystyle{16}+\sqrt{{{3}}}$$ at $$x=4$$, thus it is continuous at that point.