Limits of Compositions
Read this topicExplanations 2
- In Continuity of Compositions, we saw that f(g(x))f(g(x))f(g(x)) is continuous at ppp if g(x)g(x)g(x) is continuous at ppp and f(x)f(x)f(x) is continuous at g(p)g(p)g(p). Let's …
By Victor Wang, Moor Xu
- (Video) Limits of Composite Functions by Tom Kaster Here's a good video by Tom Kaster explaining how the limits of compositions work.
By Alex Federspiel
Problems 8
- Let fff and ggg be continuous functions. If limx→2f(x)=1\lim_{x \to 2} f(x) = 1limx→2f(x)=1, limx→1f(x)=2\lim_{x \to 1} f(x) = 2limx→1f(x)=2, limx→0g(x)=2\lim_{x \to 0} g(x…
- Let f(x)=xf(x)=\sqrt{x}f(x)=x and g(x)=x2+3x+11g(x)=x^2+3x+11g(x)=x2+3x+11. Find limx→−1(f∘g)\lim_{x\to-1}{(f\circ g)}limx→−1(f∘g).
- True or False\text{True or False}True or False === limx→c(f∘g)=limx→cf(x)∗limx→cg(x)\lim_{x\to c}{(f\circ g)}=\lim_{x\to c}{f(x)}*\lim_{x\to c}{g(x)}limx→c(f∘g)=limx→cf(x)∗…
- Let f(x)=x2+1f(x)=x^2+1f(x)=x2+1 and g(x)=exg(x)=e^xg(x)=ex and h(x)=1xh(x)=\frac1xh(x)=x1. Find limx→0(h∘g∘f)\lim_{x\to0}{(h\circ g\circ f)}limx→0(h∘g∘f).
- Let f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=exg(x)=e^xg(x)=ex and h(x)=1xh(x)=\frac1xh(x)=x1. Find limx→0(h∘g∘f)\lim_{x\to0}{(h\circ g\circ f)}limx→0(h∘g∘f).
- Let f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=exg(x)=e^xg(x)=ex and h(x)=1xh(x)=\frac1xh(x)=x1. Find limx→0(g∘h∘f)\lim_{x\to0}{(g\circ h\circ f)}limx→0(g∘h∘f).
- Let f(x)=x2−1f(x)=x^2-1f(x)=x2−1 and g(x)=exg(x)=e^xg(x)=ex and h(x)=1xh(x)=\frac1xh(x)=x1. Find limx→0(f∘g∘h)\lim_{x\to0}{(f\circ g\circ h)}limx→0(f∘g∘h).
- Let f(x)=sin xxf(x)=\dfrac{\sin~x}{x}f(x)=xsin x, and g(x)=x2+1g(x)=x^2+1g(x)=x2+1. Find the value of limx→0g(f(x))\lim_{x\rightarrow0} g(f(x))x→0limg(f(x)).