Precalculus Concepts: What Is a Limit?
Read this topicExplanations 5
- Limits are an important concept in calculus, but are actually fairly easy to understand. A limit looks something like the following: limx→0f(x)\lim\limits_{x\to 0}f(x)x→0limf(…
By Ruiran Xun
- We have discussed limit notation and rules already. If you missed that, click on their links to learn about those before learning to evaluate! Limits are used often in precalcul…
By Amy Feraco, Joshua Siktar, Victor Wang, Alex Federspiel
- When we evaluate a function at a point, say calculating f(a)f(a)f(a) for a function fff and a real number a,a,a, we gain information about the behavior of the function at that p…
By Joshua Siktar
- Limit Notation Limits are used often in precalculus and calculus. They are an important concept to understand, and they are fairly simple. First, we will discuss the notation of…
By Amy Feraco, Victor Wang
- Limit Rules or Laws We have discussed limit notation, now let's discuss some limit rules. Before we start evaluating limits, here are some laws of limits that will help with per…
By Amy Feraco, Victor Wang
Problems 27
- What is the value of limx→2f(x)\lim\limits_{x\to 2}f(x)x→2limf(x)?
- What is the value of limx→+∞f(x)\lim\limits_{x\to +\infty}f(x)x→+∞limf(x)?
- What is the value of limx→0+f(x)\lim\limits_{x\to 0^+}f(x)x→0+limf(x)?
- What is the value of limx→0−f(x)\lim\limits_{x\to 0^-}f(x)x→0−limf(x)?
- What is the value of limx→1f(x)\lim\limits_{x\to 1}f(x)x→1limf(x)?
- What is the value of limx→2(f(x)+g(x))\lim\limits_{x\to 2}\left(f(x) + g(x)\right)x→2lim(f(x)+g(x))?
- Find the following limit using algebra to simplify the equation. limx→0(4+x)−2−4−2x\lim_{x\to0} \frac{(4+x)^{-2}-4^{-2}}{x}x→0limx(4+x)−2−4−2
- Evaluate the following limit. limx→−32x2+6xx2+5x+6\lim_{x\to-3} \frac{2x^2+6x}{x^2+5x+6}x→−3limx2+5x+62x2+6x
- Evaluate the following limit. limx→11−x1−x\lim_{x\to1} \frac{1-\sqrt{x}}{1-x}x→1lim1−x1−x
- Evaluate the following limit. limx→1x2+1x−1\lim_{x\to1} \frac{x^2+1}{x-1}x→1limx−1x2+1
- If ccc is a constant, what is the following limit? limx→ac\lim_{x\to{a}} cx→alimc
- True or False\text{True or False}True or False === If the limits limx→af(x)\lim_{x\to{a}} f(x)limx→af(x) and limx→ag(x)\lim_{x\to{a}} g(x)limx→ag(x) exist, then limx→af(x)−…
- Evaluate the following limit. limx→1((1+x)x)2\lim_{x\to1} \left(\frac{(1+x)}{x}\right)^2x→1lim(x(1+x))2
- If limx→ag(x)=5\lim_{x\to{a}} g(x)=5limx→ag(x)=5, and f(x)=x+1f(x)=x+1f(x)=x+1, what is the limx→af(g(x))\lim_{x\to{a}} f(g(x))limx→af(g(x))?
- Given f(x)=12x+3f(x)=\frac{1}{2}x+3f(x)=21x+3 and g(x)=x+1g(x)=x+1g(x)=x+1, what is limx→2f(x)⋅g(x)\lim_{x\to2} f(x) \cdot {g(x)}limx→2f(x)⋅g(x)?
- Given f(x)=12x+3f(x)=\frac{1}{2}x+3f(x)=21x+3 and g(x)=x+1g(x)=x+1g(x)=x+1, what is limx→4f(x)g(x)\lim_{x\to4} \frac{f(x)}{g(x)}limx→4g(x)f(x)?
- Given f(x)=23x+4f(x)=\frac{2}{3}x+4f(x)=32x+4, what is limx→6f(x)3\lim_{x\to6} \sqrt[3]{f(x)}limx→63f(x)?
- Which of the following roughly define 0−?0^{-}?0−?
- Which of the following are indeterminate terms?
- True or False\text{True or False}True or False === The limit does not exist if limx→a−f(x)=limx→a+f(x)\lim_{x\to{a^{-}}}f(x) = \lim_{x\to{a^{+}}}f(x)x→a−limf(x)=x→a+limf(x)
- Which of the following is the first step when evaluating any limit?
- Evaluate limx→1212x+74\lim_{x\rightarrow{\frac{1}{2}}}\frac{1}{2}x+\frac{7}{4}limx→2121x+47.
- What is the limit as xxx approaches 444 of the graph above?
- What is the limit as xxx approaches −4-4−4 in the graph above?
- Evaluate the limit. limx→∞212\lim_{x\to\infty} 212x→∞lim212
- Evaluate the following limit. limx→−75x2+32x−21x2−49\lim_{x\to-7} \frac{5x^2+32x-21}{x^2-49}x→−7limx2−495x2+32x−21
- Let λ∈(0,1).\lambda \in (0, 1).λ∈(0,1). Find the value of limx→∞ln(xλ(x+1)λ−1)λx+(1−λ)(x+1).\lim_{x \rightarrow \infty}\frac{\ln\left(\frac{x^{\lambda}}{(x + 1)^{\lambda - 1}}…