Limits – All You Need To Know

Limits is all about the output function, if x is really close to a limiting number, infinity or zero.
Limits are used when when we want to evaluate a function at a particular value, but can’t substitute the value in straight because the formula “blows up”. You have to do something to the formula first.

There are few types of limits….

Type 1: In first type we take an equation with a variable x, and see what number it approaches at it reaches infinity. A simple one is 1/x. As x gets closer to infinity, it gets closer and closer to a value of zero, but it will never actually be zero.


Type 2: The other type of limit is the limit of a sum or integral of an equation as x goes from a to b where either a or b or both are infinity. Obviously something like the limit of the sum of 1 + x from 0 to infinity does not have a limit as 1+x keeps getting larger. But some equations do decrease as x gets higher than value and some of those would decrease enough that the upper limit of that sum will approach a value, but never reach it.

Basic Facts

When does a formula “Blow Up”?

X/0, when you have to divide a value from zero.
0/0, when you have to divide zero by zero.

Sides of Limits

There are two sides of limits known as left-hand limits and right-hand limits.

When does a limit not exist?

  • When the left-and right-hand limits aren’t equal (resulting in a discontinuity in the function).
  • When  a function increases or decreases infinitely (“without bound”) as it approaches a given x-value.
  • In the cases of infinite oscillation when approaching a fixed point

Approaching Infinity

1/∞ is undefined. But we can still approach to a value.

<math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo> </mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math>“></p>



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<h2><strong>Evaluating methods for Limits (E-value-ating)</strong></h2>



<ol><li>Substitution Method</li><li>Factorizen Method</li><li>Conjugate Method</li><li>Limits of Rational Functions</li><li>L’Hôpital’s Rule</li><li>Formal Method</li></ol>



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<h3><strong>Substitution Method – Limits Evaluating</strong></h3>



<p>In substitution method, we put the value of the limit to the equation and check whether it gets a valid answer.</p>



<p><img loading= Categories Engineering Mathematics