# 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.