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Problems and Solutions of Basic Limits: Here we will discuss various basic problems of limits with solutions. Example 1: Evaluate $\lim\limits_{x \to 0} (\sin x + \cos x)$ Solution:      $\lim\limits_{x \to 0} (\sin x +\cos x)$ $=\lim\limits_{x \to 0} \sin x + \lim\limits_{x \to 0}\cos x$ $=\sin 0 + \cos 0$ $=0+1$ $=1.$ 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