*To do the hard limit that we want, $\lim_ (\sin x)/x$, we will find two simpler functions $g$ and $h$ so that $g(x)\le (\sin x)/x\le h(x)$, and so that $\lim_g(x)=\lim_h(x)$.*

With a little algebra this turns into $(\sin x)/x \le 1/\cos x$, giving us the $h$ we seek.

To find $g$, we note that the circular wedge is completely contained inside the larger triangle.

So now we have $$ \cos x \le \le .$$ Finally, the two limits $\lim_\cos x$ and $\lim_1/\cos x$ are easy, because $\cos(0)=1$.

By the squeeze theorem, $\lim_ (\sin x)/x = 1$ as well.

This formula is not applicable when n is even ► Sometimes, we can directly use below formula to evaluate the limit ►If the degree of the numerator is more than or equal to the degree of the denominator, then we should divide.

Example-2:- Evaluate Ans:- = 10 Rationalization Method:- If we ever get 0/0 form in the problems involving square roots, then there must be a common factor in both numerator and denominator which must be canceled out to get a meaningful form.

To do this we need to be quite clever, and to employ some indirect reasoning.

The indirect reasoning is embodied in a theorem, frequently called the squeeze theorem.

If f(x) and g(x) be two functions in such a way that (i) or (ii) both are continuous at x=a (iii) both are differentiable at x=a (iv) f'(x) and g'(x) are both continuous at x=a then If the result is still in 0/0 or in the form we can again differentiate and write like this You can download this PDF to get Practice Questions.

We want to compute this limit: $$\lim_ .$$ Equivalently, to make the notation a bit simpler, we can compute $$\lim_ .$$ In the original context we need to keep $x$ and $\Delta x$ separate, but here it doesn't hurt to rename $\Delta x$ to something more convenient.

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## Solving limit problems - uk

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## Limits as x Approaches a Constant - UC Davis Mathematics

The following problems require the use of the algebraic computation of limits of functions as x approaches a constant. Most problems are average. A few are somewhat challenging. All of the solutions are given WITHOUT the use of L'Hopital's Rule. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by giving careful consideration to the form.…

## Solving Limits at Infinity Intuition and Examples

Solving Limits at Infinity The neat thing about limits at infinity is that using a single technique you'll be able to solve almost any limit of this type. In the following video I go through the technique and I show one example using the technique.…

## Calculus Solving Limits

In this lesson we'll solve limits analytically. We'll see examples of solving limits with rationalization, solving limits by factoring, and solving limits of complex fractions.…

## Limit and Continuity Graphs Practice Questions - dummies

In addition to solving limit problems numerically with your calculator and symbolically with algebra, you should be able to solve limit and continuity problems visually. The following practice questions will test your skills. Use the following figure to answer the practice problems. Practice questions f5 =…

## Calculus I - Computing Limits

There is not really a lot to this problem. Simply recall the basic ideas for computing limits that we looked at in this section. We know that the first thing that we should try to do is simply plug in the value and see if we can compute the limit.…

## Solve limit problems - Expert Link

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