Question

Find the limits. \(f(x)=5-x, g(x)=x^{3}\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 4} g(x)\) (c) \(\lim _{x \rightarrow 1} g(f(x))\)

Short answer

The solutions are: (a) \(\lim _{x \rightarrow 1} f(x) = 4\), (b) \(\lim _{x \rightarrow 4} g(x) = 64\), and (c) \(\lim _{x \rightarrow 1} g(f(x)) = 64\).

Step-by-step solution

Find \(\lim _{x \rightarrow 1} f(x)\)

Substitute 'x = 1' directly into the function \(f(x)=5-x\).\nSo, \[\lim _{x \rightarrow 1} f(x)=f(1)=5-1=4.\]

Find \(\lim _{x \rightarrow 4} g(x)\)

Substitute 'x = 4' directly into the function \(g(x)=x^{3}\).\nThis gives \[\lim _{x \rightarrow 4} g(x)=g(4)=4^{3}=64.\]

Find \(\lim _{x \rightarrow 1} g(f(x))\)

First we substitute \(f(x)\) into \(g(x)\) to create a new function, let's call this function \(h(x)\). Hence, \(h(x) = g(f(x)) = (5 - x)^3\). Then we substitute 'x = 1' into \(h(x)\). This gives, \[\lim _{x \rightarrow 1} g(f(x))=h(1)=(5 - 1)^3=4^{3}=64.\]