The sum rule of limits says that the limit of the sum of two functions is the same as the sum of the limits of the individual functions. In this post, we will prove the sum/addition rule of limits by the epsilon-delta method. Question: What is an epsilon-delta proof of the sum law of … Read more
The epsilon-delta definition of limit says that if limx→a f(x) = L, then for every ε>0, there exists a δ>0 such that |f(x)-L| < ε whenever 0<|x-a|<δ. In this post, we will learn the epsilon-delta definition of a limit with examples. We also provide the negative statement of the epsilon-delta definition of limits. Epsilon-Delta Definition of … Read more
Trigonometric Limits: Problems and Solutions We will compute the limits of trigonometric functions. A few examples of trigonometric functions are `sin x, cos x, tan x, sin (x+a), frac{sin x}{x}` etc. At first, recall the well-known formulas of trigonometric limits. 1. `lim_{x to 0} sin x =0` 2. `lim_{x to 0} cos x =1` 3. `lim_{x to 0} … Read more
Problems and Solutions of Basic Limits: Here we will discuss various basic problems of limits with solutions. Example 1: Evaluate `lim_{x to 0} (sin x + cos x)` Solution: `lim_{x to 0} (sin x +cos x)` `=lim_{x to 0} sin x + lim_{x to 0}cos x` `=sin 0 + cos 0` `=0+1` `=1.` Example 2: Evaluate … Read more
First, recall the main formulas in the context of exponential/logarithmic limits. They will be widely used in the problems below: Formula 1. $\lim\limits_{x \to 0} \dfrac{e^x-1}{x}=1$ Formula 2. $\lim\limits_{x \to 0} \dfrac{\log_e(1+x)}{x}=1$ Formula 3. $\lim\limits_{x \to 0} \dfrac{a^x-1}{x}=\log_e a$ Exponential Logarithmic Limits: Problems and Solutions Problem 1: $\lim\limits_{x \to 0} \dfrac{e^{ax}-1}{x}$ Solution: Put $z=ax.$ Then … Read more
Limit Formulas with Their Proofs: The limit of a function is an important concept in the theory of Calculus. In this section, we will prove all the limit formulas. Formula 1: `lim_{x to 0} frac{sin x}{x}=1` Formula 2: `lim_{x to a} frac{x^n-a^n}{x-a}=na^{n-1}` Formula 3: `lim_{x to 0} frac{e^x-1}{x}=1` Formula 4: `lim_{x to 0} frac{log_e(1+x)}{x}=1` Formula 5: `lim_{x to 0} … Read more
Let us note down the list of all limit formulas. Here we will list all the important limit formulas and see how to apply such formulas in practical examples. Limit Formulas with Properties and Applications Notation and Definition: Let f(x) be a function of x. The limit of f(x) as x goes to a (x→a) … Read more