Question

Evaluate the following limit.
$\mathop {\lim }\limits_{x \to 1} \dfrac{{{x^2} + 1}}{{x + 1}}$

Answer 1

Hint: Here we need to simplify the limit and then substitute the value of x to evaluate the limit.

Complete step-by-step answer:
Given,
$\mathop {\lim }\limits_{x \to 1} \dfrac{{{x^2} + 1}}{{x + 1}}$
Add and subtract 2x in the numerator we get
$
  \mathop {\lim }\limits_{x \to 1} \dfrac{{{x^2} + 1 + 2x - 2x}}{{x + 1}} \\
  \mathop {\lim }\limits_{x \to 1} \dfrac{{{{(x + 1)}^2} - 2x}}{{x + 1}} \\
  \mathop {\lim }\limits_{x \to 1} \dfrac{{{{(x + 1)}^2}}}{{x + 1}} - \mathop {\lim }\limits_{x \to 1} \dfrac{{2x}}{{x + 1}} \\
  \mathop {\lim }\limits_{x \to 1} (x + 1) - \mathop {\lim }\limits_{x \to 1} \dfrac{{2x}}{{x + 1}} \\
$
Now put the value of x we get
$(1 + 1) - \dfrac{{2 \times 1}}{{1 + 1}} \Rightarrow 2 - \dfrac{2}{2} \Rightarrow 2 - 1 = 1$
Answer is 1.

Note: We can solve these questions by just putting the value of x but sometimes we have to perform some operation (addition and subtraction) on the denominator or numerator to solve the problem.