Direct Substitution Method to Solve Limits

Here you will learn direct substitution method to solve limits with examples.

Let’s begin –

Direct Substitution Method to Solve Limits

Consider the following limits :

(i)  \(lim_{x \to a}\) f(x)

(ii)  \(lim_{x \to a} {\phi(x)\over \psi(x)}\)

If f(a) and \(\phi(a)\over \psi(a)\) exist and are fixed real numbers, then we say that

\(lim_{x \to a}\) f(x) = f(a)  and  \(lim_{x \to a} {\phi(x)\over \psi(x)}\) = \(\phi(a)\over \psi(a)\)

In other words, if the direct substitution of the point, to which the variable tends to, we obtain a fixed real number, then the number obtained is the limit of the function.

Infact, if the point to which the variable tends to is a point in the domain of the function, then the value of the function at that point is its limit.

Also Read : How to Solve Indeterminate Forms of Limits

Following examples will illustrate the above method.

Example : Evaluate : \(lim_{x \to 1}\) \(3x^2 + 4x + 5\).

Solution : We have,

\(lim_{x \to 1}\) \(3x^2 + 4x + 5\) = \(3(1)^2 + 4(1) + 5\) = 12.

Example : Evaluate : \(lim_{x \to 2}\) \(x^2 – 4\over x + 3\).

Solution : We have,

\(lim_{x \to 2}\) \(x^2 – 4\over x + 3\) = \(4 – 4\over 2 + 3\) = 0.

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